Exact and Approximate Algorithms for New Variants of Some Classic

Transcription

Tel Aviv University
The Raymond and Beverly Sackler Faculty of Exact Sciences
The Blavatnik School of Computer Science
Exact and Approximate Algorithms
for New Variants of Some Classic
Graph Problems
Thesis submitted for the degree of Doctor of Philosophy by
Amitai Armon
Under the supervision of Prof. Uri Zwick
Submitted to the Senate of Tel-Aviv University
September 2008
To my mother and to my grandmother Belina
Acknowledgment
To my advisor Uri Zwick, for his successful guidance of this research, and for sharing with
me his optimistic approach to settling algorithmic problems.
To Amos Fiat, Micha Sharir, Vera Asodi, Eyal Even-Dar and Ben Sandbank, for long
collaborations in fundamental course teaching, which were very pleasant.
To Adi Avidor and Oded Schwartz, for a fruitful research collaboration and a joyful joint
work.
To Amir Epstein and Ido Tzameret, for being great partners to share an office with.
And to my mother and my grandmother Belina, for all the love and support, and for making
me who I am.
Abstract
Graphs are probably the most studied object in theoretical computer-science, and many graph
problems, such as Shortest-Paths, Max-Flow and the Traveling Salesperson Problem, have been
studied for many decades. Efficient algorithms have been developed for some of these problems,
while others were proven to be hard to solve. For many of the latter problems, efficient algorithms
which find an approximate solution have been developed.
In this work we consider new variants of three well-studied fundamental graph problems:
Global Min-Cut, The Traveling Salesperson Problem and Facility Location. For each of these
problems, the variants we consider generalize or extend the respective problem, by considering
additional goals, options or constraints, arising either from real-life scenarios or from previous
theoretic studies. In our study of these new variants, we also answer some open questions posed
regarding previously introduced special cases, and improve some previous results.
In Chapter 2 we consider multicriteria versions of the Global Min-Cut problem. In the kcriteria setting of Global Min-Cut, each edge of the input graph has k non-negative costs associated
with it. These costs are measured in separate, non interchangeable, units. In the AND-version
of the problem, purchasing an edge requires the payment of all the k costs associated with it.
In the OR-version, an edge can be purchased by paying any one of the k-costs associated with
it. Given k bounds b1 , b2 , . . . , bk , the basic multicriteria decision problem is whether there exists
a cut C of the graph that can be purchased using a budget of bi units of the i-th criterion, for
1 ≤ i ≤ k.
We show that the AND-version can be solved in polynomial-time for any fixed number k
of criteria, and it is NP-hard for non-fixed k. Our results may be somewhat surprising, since
Papadimitriou and Yannakakis [PY00] proved that bicriteria s-t-Min-Cut is strongly NP-hard.
Our work resolves an open question of Bruglieri et al. [BEH00, BME04], who asked whether a
cardinality-constrained variant of Global Min-Cut, in which the number of edges connecting the
two subsets is limited by some input number, can be solved in polynomial-time. We answer their
question in the affirmative.
Regarding the OR-version of the problem, on the other hand, we show NP-hardness even
for k = 2, and prove that the problem can be solved in pseudo-polynomial time for any fixed
number k of criteria. It also admits an FPTAS (a fully-polynomial-time approximation scheme).
Further extensions and applications, as well as multicriteria OR-versions of two other optimization
problems, are also considered. As far as we know, we are the first to consider OR-versions of
multicriteria problems. This chapter is based on the paper [AZ06].
In Chapter 3 we consider cooperative variants of The Traveling Salesperson Problem (TSP).
In these problems a salesperson has to make deliveries to customers who are willing to help in the
process. The basic motivation for these variants is that in many realistic scenarios the “customers”
are actually other members of the same organization/company the salesperson belongs to, and
thus can be asked to help. The customers may be able to cooperate in several modes: They
may assist by approaching the salesperson to receive the goods, by delivering goods that they
received to other customers, or by doing both. Several objectives may be of interest: Minimizing
the total distance traveled by all the participants, minimizing the maximal distance traveled by
a participant, or minimizing the total time until all the deliveries are made.
All the combinations of cooperation-modes and objective functions are considered in our study,
both in weighted undirected graphs and in Euclidean space. We show that most of the problems
we consider have a constant approximation algorithm, many of the others admit a PTAS, and a
few are solvable in polynomial time. On the intractability side, we provide NP-hardness proofs and
inapproximability factors, some of which are tight. All our algorithms are purely combinatorial,
and our hardness proofs use reductions from well-known NP-hard problems, without requiring
the use of the PCP theorem. This chapter is based on the paper [AAS06].
Chapter 4 considers a min-max version of the previously studied r-gathering problem with
unit-demands. The problem we consider is a metric facility-location problem, in which each open
facility must serve at least r customers, and the maximum of all the facility and connection costs
should be minimized (rather than their sum). This problem is motivated by scenarios in which r
customers are required for a facility to be worth opening, and the costs represent the time until
the facility/connection will be available (i.e., we want to have the complete solution ready as soon
as possible).
We present a 3-approximation algorithm for this problem, and prove that it cannot be approximated better (assuming P 6= N P ). Next we consider this problem with the additional
natural requirement that each customer will be assigned to a nearest open facility, and present a
9-approximation algorithm. We further consider previously introduced special cases and variants,
and obtain improved algorithmic and hardness results. The results of this chapter are based on
the paper [Arm08].
Contents
1 Introduction
1.1
1.2
1.3
1.4
1
Multicriteria Global Min-Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.1.1
Our Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Cooperative TSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2.1
4
Min-Max r-Gatherings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3.1
Our contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
General Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.4.1
Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.4.2
Gap-Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2 Multicriteria Global Min-Cut
9
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2
Multicriteria global minimum cut: the AND-version . . . . . . . . . . . . . . . . .
12
2.2.1
The Min-Max problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.2.2
The decision problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.2.3
The optimization problem . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.2.4
Two applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
2.2.5
The Pareto Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2.2.6
Hardness results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
Multicriteria global minimum cut: the OR-version . . . . . . . . . . . . . . . . . .
18
2.3.1
Relation to scheduling on unrelated machines . . . . . . . . . . . . . . . . .
18
2.3.2
The min-max version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.3.3
A case that can be solved in polynomial time . . . . . . . . . . . . . . . . .
20
OR-versions of other multicriteria problems . . . . . . . . . . . . . . . . . . . . . .
20
2.4.1
21
2.3
2.4
Shortest paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i
2.4.2
2.5
Minimum spanning trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
3 Cooperative TSP
3.1
3.2
3.3
3.4
23
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.1.1
Related Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.1.2
26
Euclidean cTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.2.1
Min-Sum Euclidean-cTSP . . . . . . . . . . . . . . . . . . . . . . . . . .
27
3.2.2
Min-Max Euclidean-cTSP . . . . . . . . . . . . . . . . . . . . . . . . . .
36
3.2.3
Min-Makespan Euclidean-cTSP . . . . . . . . . . . . . . . . . . . . . .
38
cTSP in Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.3.1
Min-Sum cTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.3.2
Min-Max cTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
3.3.3
Min-Makespan cTSP . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
Discussion and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4 On unweighted r-Gatherings
57
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
4.1.1
Our results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
4.2
Problem Definitions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
4.3
Approximating Min-Max r-Gathering . . . . . . . . . . . . . . . . . . . . . . . . .
61
4.4
Assigning to a Nearest Open Facility . . . . . . . . . . . . . . . . . . . . . . . . . .
64
4.4.1
Improved Results for r = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
4.5
Hardness Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
4.6
Concluding Remarks and Open Problems . . . . . . . . . . . . . . . . . . . . . . .
72
4.1
5 Concluding Remarks
75
Bibliography
77
ii
Chapter 1
Introduction
Graphs are apparently the most studied object in theoretical Computer-Science, and many of the
graph problems have been studied for many decades. A partial list of such problems includes the
well-known Shortest-Paths problem, Minimum Spanning Tree, Max-Flow, Global Min-Cut, The
Traveling Salesperson Problem, Minimum Steiner Tree, k-Center, k-Median, Facility-Location,
and many more. A basic survey can be found in the fundamental textbook of Cormen et al.
[CLRS01]. Efficient algorithms have been developed for some of these classical problems, while
other problems were proven to be hard to solve. For many of the latter problems, efficient
algorithms which find an approximate solution have been developed (see, e.g, [Vaz03] for a survey
of many of these results).
In parallel to the on-going study of classical graph problems, there are many studies on their
special cases, generalizations and other variants. Special cases often relate to special graphs or to
limiting some input values (e.g limiting them to integers or to a certain range). Generalizations
sometimes involve, for example, considering directed rather than undirected graphs, or considering matroids rather than graphs. Other types of variants involve changing the goal function
(sum/max/min/etc.), adding/removing a constraint, and/or considering a different space (e.g.
the plane or another Euclidean space). Such variants were sometimes motivated by theoretic
questions and sometimes evolved from real-life applications.
In this work we consider new variants of three well-studied fundamental graph problems: MinCut, The Traveling Salesperson Problem and Facility Location. For each of these problems, the
variants we consider seem to provide a new insight on the respective problem, and include as
special cases previously introduced variants. Our work answers some open questions posed regarding these special cases, improves some previous results, and expands the scope of the previous
research.
2
Introduction
1.1
Multicriteria Global Min-Cut
In Chapter 2 we consider multicriteria versions of the well-known Global Min-Cut problem. In
Global Min-Cut, the vertices of an edge-weighted undirected graph should be split into two nonempty subsets. The goal is that the total weight (cost) of the edges connecting the two subsets will
be minimal. Karger provided a near-linear (O(m log3 n)) Monte-Carlo algorithm for this problem
[Kar00]. The fastest deterministic algorithms require O(mn log n) time [HO94, NI92, SW97],
similarly to the fastest max-flow algorithms which solve the s-t-Min-Cut problem - the variant in
which two pre-specified vertices must be in different subsets (see, e.g., [GT88]).
Bruglieri et al. [BEH00, BME04] asked whether a cardinality-constrained variant of this
problem, in which the number of edges connecting the two subsets is limited by some input
number, can be solved in polynomial-time. This problem can be viewed as a special case of a
bicriteria variant of Global Min-Cut, in which each edge has an additional type of weight (equal
to 1 for all the edges in this special case), and the minimization of the first weight (criterion)
should be done without exceeding the limit for the second weight (which is not interchangeable
with the first one).
There is a large body of works, mostly by the Operations Research community, on cardinalityconstrained and multicriteria problems, which are usually NP-hard and require approximations
(see, e.g., [EG00, Ehr00, Cli97]). Specifically related to this problem is the result of Papadimitriou
and Yannakakis [PY00], who proved that bicriteria s-t-Min-Cut is strongly NP-hard.
1.1.1
Our Contribution
Quite surprisingly, we show that bicriteria Global Min-Cut can be solved in polynomial time.
Thus we also resolve the question asked by Bruglieri et al.
We extend our study by considering two more general multicriteria versions of Global MinCut. In the k-criteria setting, each edge of the input graph has k non-negative costs associated
with it. These costs are measured in separate, non-interchangeable, units. In the AND-version
of the problem, purchasing an edge requires the payment of all the k costs associated with it.
In the OR-version, an edge can be purchased by paying any one of the k-costs associated with
it. Given k bounds b1 , b2 , . . . , bk , the basic multicriteria decision problem is whether there exists
a cut C of the graph that can be purchased using a budget of bi units of the i-th criterion, for
1 ≤ i ≤ k.
We show that the AND-version is in P for any fixed number k of criteria, and is NP-hard
for non-fixed k. The OR-version of the problem, on the other hand, is NP-hard even for k = 2,
but can be solved in pseudo-polynomial time for any fixed number k of criteria. It also admits
1.2 Cooperative TSP
3
an FPTAS (a fully-polynomial-time approximation scheme). We provide similar results for the
optimization versions of these two problems (minimizing the cost in one criterion given bounds
on the others). Further extensions and applications, as well as multicriteria OR-versions of two
other optimization problems, are also considered. As far as we know, we are the first to consider
OR-versions of multicriteria problems. This chapter is based on the paper [AZ06].
1.2
Cooperative TSP
In Chapter 3 we consider cooperative variants of The Traveling Salesperson Problem (TSP).
The input to TSP consists of a complete edge-weighted undirected graph, and the goal is to
find a cycle in which each vertex appears exactly once, such that its total weight is minimal. The
classical motivation is planning a tour for a salesperson, wishing to visit a set of cities/customers
and return home, while minimizing the travel. This problem is known to be NP-hard [GGJ76],
and cannot be approximated within any polynomial factor [Vaz03]. For the metric version, in
which the weights satisfy the triangle inequality, Christofides presented a 3/2 approximation
algorithm [Chr76], and this variant cannot be approximated within less than a factor of
131
130 ,
assuming P 6= N P [EK01]. In any fixed-dimension Euclidean space, the problem has a PTAS (a
polynomial-time approximation scheme) [Aro98, Mit99], and is still NP-hard [Pap77]. There is
also a 3/2 approximation algorithm for path-TSP [Hoo91], the variant in which the salesperson
does not need to return home (i.e., one needs to find a path instead of a cycle, and one endpoint
of the path is specified in the input).
TSP has been studied in many forms over the decades. One of the interesting recent variants
of TSP is the Freeze-Tag problem [ABF+ 02, SABM02, ABG+ 03, KLS04], presented by Arkin et
al. [ABF+ 02], which is motivated by a swarm of robots scenario. In that scenario, an awake
robot has to wake up a set of sleeping robots, and when a robot is awaken it can be instructed
how to move in order to wake up other robots. The goal is to finish waking up all the robots
as fast as possible (i.e., minimizing the “makespan”). Arkin et al. presented constant factor
approximation algorithms for this problem on some special graphs, and proved that it cannot
be approximated within less than a factor of 5/3 (assuming P 6= N P ). Konemann et al. later
√
presented an O( log n) approximation algorithm for the problem on general graphs [KLS04].
The Freeze-Tag problem can be viewed as a variant of TSP, in which the “customers” cooperate
with the salesperson, by helping in the “sales” after they are reached. Such cooperativeness can
occur in various other realistic scenarios, in which the customers are in fact “agents” of the same
organization as the salesperson, and can be instructed to move in order to assist in “delivering
the goods”. Clearly, there can be other forms of such a cooperation. For example, the customers
4
Introduction
might be able to move before they receive the goods, in order to approach the salesperson. In
some scenarios, this type of cooperation might be relevant instead of (or in addition to) moving
after receiving the goods. Also, various goals other than minimizing the makespan might be of
interest. For example, in the sleeping robots scenario, if each robot has limited battery, then it
might be interesting to minimize the maximal travel of any of the robots. Another interesting
goal in some scenarios might be minimizing the total travel of all the participants (e.g. if their
company pays all their travel costs). Clearly, just like in classical TSP, it might be appropriate
to give up the requirement that the salesperson and customers will “return home” (their original
location might be arbitrary). Also, such “cooperative” variants can be considered in a Euclidean
space rather than a graph.
1.2.1
Our Contribution
As described in Chapter 3, we studied all the above mentioned combinations of cooperationmode, goal-function, space and “roundtrip requirement”. We show that most of these problems
have a constant-factor approximation algorithm, many of the others admit a PTAS, and a few
are solvable in polynomial time. On the intractability side we provide NP-hardness proofs and
inapproximability factors, some of which are tight. All our algorithms are purely combinatorial,
and our hardness proofs use reductions from well-known NP-hard problems, without requiring
the use of the PCP theorem. This chapter is based on the paper [AAS06].
1.3
Min-Max r-Gatherings
Chapter 4 considers the r-gathering problem, a recently introduced variant of the Facility Location
problem. In the classical Facility Location, the input consists of a set of customers C and a set
of potential facility locations F . Each potential location is associated with a cost for opening a
facility there. Each pair of customer location and facility location is associated with a servicecost (for serving that customer by that facility). The problem is to choose locations for opening
facilities, such that the sum of opening costs and service costs is minimal (each customer is served
by the open facility that minimizes his service cost). This problem is often described by a bipartite
graph, whose sides correspond to C and F (associated with costs) and whose edge weights are
the respective service costs.
Hochbaum [Hoc82] presented an O(log n) approximation for this problem. The problem cannot
be approximated within an o(log n) factor [Arc00], unless N P ⊆ DT IM E(nO(log log n) ). If the
service costs are assumed to satisfy the triangle-inequality, then the problem can be approximated
1.3 Min-Max r-Gatherings
5
within a factor of 1.5 [Byr07], and cannot be approximated within less than a factor of 1.463, unless
N P ⊆ DT IM E(nO(log log n) ) [GK99]. Most of the research deals with the scenario in which the
service-costs satisfy the triangle-inequality, called metric facility location. The costs are sometimes
called distances in this scenario, and thus each customer is assigned to the nearest open facility.
This problem has been extensively studied for many decades and has many well-studied variants
(see, e.g., [Dre95, Vyg05] for surveys).
Some of the more recent variants of Facility Location consider an additional constraint on
the number of customers in each facility. In the capacitated variant of the problem (see, e.g.,
[Vyg05]), there is an upper bound on the number of customers that each facility may serve.
In the soft-capacities version, more than one facility may be opened at the same location (the
location’s opening cost is thus multiplied by the number of facilities opened there). In the hard
capacities version, only one facility may be opened in each location. For soft capacities, there
is a 2-approximation algorithm, which matches the integrality-gap of the linear relaxation of
the problem [MYZ03]. For hard capacities, there is a 5.83-approximation algorithm [ZCY04].
The best lower bound is the lower bound of 1.463 for the uncapacitated version (unless N P ⊆
DT IM E(nO(log log n) )) [ZCY04, GK99]. Note that unlike the classical facility location problem,
a customer might not be served by the facility with minimal service cost, due to the constraint
posed by the capacities.
One of the more recent variants of Facility Location that consider the number of customers
served by each facility is the r-gathering problem [GMM00, KM00, Svi08]. This problem is an
analogue of capacitated Facility Location, in which there is a lower bound of r on the number of
customers served by each facility (rather than an upper bound). The basic motivation is that a
certain number of customers is usually required to make a facility worth opening. This problem
can also be regarded as a clustering problem (without opening costs), in which small clusters are
not desired (see [AFK+ 06]). Previous works also considered the case in which the lower bounds
differ between different locations [GMM00, KM00]. Again, a customer might not be served by
the facility with minimal service cost, due to the constraint posed by the lower-bounds.
Karger and Minkoff [KM00] and Guha et al. [GMM00], who introduced the r-gathering
problem, also provided a bicriteria approximation algorithm for it. Their algorithm guarantees
that each facility will serve at least αr customers, at a cost of at most
1+α
1−α β
times the optimal
cost for r-gathering, where β is the approximation ratio for Facility Location. This implies a
1.5(2r − 1 + ) single criterion approximation (choosing α =
r−1
r
+ ). Svitkina [Svi08] has recently
provided a constant (558) single-criterion approximation for the problem. The recent work of
[AK07] proves that the problem is polynomial for r = 2 if there are no facility costs.
Aggrawal et al. [AFK+ 06] considered a special case of a min-max version of this problem, in
6
Introduction
which there are no opening costs, C = F (i.e., the customer locations are the potential facility
locations), and the maximal service cost should be minimized. They called this special case,
motivated by a clustering application, r-gather clustering. They provided a 2-approximation
algorithm for this special case, and proved that it cannot be approximated better for r ≥ 7
(assuming P 6= N P ). For the generalization in which a certain fraction of the customers may be
ignored (“outlayer points”), they state that there is a 3-approximation algorithm.
1.3.1
Our contribution
We provide a 3-approximation algorithm for the general min-max version of r-gathering, and prove
that it cannot be approximated better for any r > 2 (the recent work of [AK07] proves that the
problem is polynomial for r = 2). We also prove that r-gather clustering cannot be approximated
within less than a factor of 2 for any r > 2 improving the result of [AFK+ 06]. Our algorithm
has several extensions, among them a 3-approximation for the generalization in which a certain
fraction of the customers may be ignored (thus solving a more general problem than [AFK+ 06],
with the same approximation ratio). We also consider a variant in which each customer in the
solution has to be assigned to the nearest open facility, which is a quite natural requirement. We
provide a 9-approximation algorithm for this variant.
For the original min-sum version of r-gathering, we provide a 2r approximation for the special
case in which there are no opening costs. This approximation ratio was also obtained by Lim et
al. [LWX06] in parallel to our work, using a different algorithm.
The results of this chapter are based on the paper [Arm08].
1.4 General Preliminaries
1.4
7
General Preliminaries
1.4.1
Approximations
Let O be a minimization problem, let ALG be an approximation algorithm for this problem, and
let I be an instance of O. Denote by OP T (I) the optimal solution for I, and by ALG(I) the
solution found by ALG applied to I. Denote by r the ratio of their values: r =
|ALG(I)|
|OP T (I)| .
Then
we say that ALG is a c-approximation for O if for every input I, r ≤ c.
A Polynomial Time Approximation Scheme (PTAS) for a minimization problem O, is an
approximation algorithm ALG that, given an input instance I and a parameter ε > 0, computes
a c- approximation for I in time t, where c ≤ 1 + ε and t = Polyε (|I|). That is, for every fixed
ε > 0 the running time of ALG is polynomial, but the dependency of the running time in ε is not
necessarily polynomial.
If t = Poly(|I|, 1ε ) then we say that ALG is a Fully Polynomial Time Approximation Scheme
(FPTAS).
This work focuses on minimization problems - similar definitions can be made for maximization
problems.
1.4.2
Gap-Problems
In order to prove inapproximability of an optimization problem, one usually defines a corresponding gap problem. Recall the following definition:
Definition 1.1 (Gap Problem - Minimization). Let O be a minimization problem. Gap-O- [α, β]
is the following decision problem:
Given an input instance, decide whether
• There exists a solution of cost at most α, or
• Every solution of the given instance is of cost larger than β.
If the cost of the solution resides between these values, then any output suffices.
Clearly, if Gap-O- [α, β] is N P -hard, then it is N P -hard to approximate O to within any
factor smaller than
β
α.
8
Introduction
Chapter 2
2.1
Introduction
We consider two multicriteria versions of the global minimum cut problem in undirected graphs.
Let G = (V, E) be an undirected graph, and let w1 , . . . , wk : E → R+ be k nonnegative cost (or
weight) functions defined on its edges. A cut C of G is a subset C ⊆ V such that C 6= φ and
C 6= V . The edges cut by this cut are E(C) = {(u, v) ∈ E | u ∈ C, v 6∈ C}. (As the graph is
undirected, C and V −C define the same cut.) In the AND-version of the k-criteria problem, the
i-th weight (or cost) of the cut is
i-th cost in the AND-version:
wi (C) =
X
wi (e) ,
1≤i≤k.
e∈E(C)
In the OR-version of the problem we pay only one of the costs associated with each edge e ∈ E(C)
of the cut. More specifically, we choose a function α : E(C) → {1, 2, . . . , k} which specifies which
cost is paid for each edge of the cut. The i-th cost of the cut C, with respect to the choice
function α, is then
i-th cost in the OR-version:
wi (C, α) =
X
wi (e) ,
1≤i≤k.
e∈E(C) ∧ α(e)=i
For the AND-version, the basic multicriteria global minimum cut decision problem asks, given k
cost bounds b1 , b2 , . . . , bk , is there a cut C such that wi (C) ≤ bi , for 1 ≤ i ≤ k? In the optimization
problem we are given k − 1 bounds b1 , b2 , . . . , bk−1 and asked to find a cut C for which wk (C)
is minimized, subject to the constraints wi (C) ≤ bi , for 1 ≤ i ≤ k − 1. The min-max version
10
of this problem asks for a cut C for which maxki=1 wi (C) is minimized, i.e., a cut whose largest
cost is as small as possible. The Pareto set P (G, w1 , . . . , wk ) ⊆ Rk of an instance hG, w1 , ..., wk i
is the set of cost vectors of cuts that are not dominated by the cost vector of any other cut. It
follows, therefore, that if (c01 , c02 , . . . , c0k ) is the cost vector of a cut C 0 of the graph, then there
exists a vector (c1 , c2 , . . . , ck ) ∈ P (G, w1 , . . . , wk ) such that ci ≤ c0i for 1 ≤ i ≤ k. Corresponding
definitions can be made for the OR-version of the problem, where α should then also be chosen.
Figure 2.1: A simple illustration for bicriteria min-cut. In the AND-version, the cost vector
of the depicted cut is (4,9). In the OR-version, its cost with the chosen α is (1,2).
Multicriteria optimization is an active field of research (see, e.g., the books of Climacao [Cli97]
and Ehrgott [Ehr00]). (Most research is focused, in our terminology, on AND-versions of various
optimization problems. All results cited below refer to the AND-versions of the problems, unless
stated otherwise.) Papadimitriou and Yannakakis [PY00] investigated the complexity of several
multicriteria optimization problems. In particular, they considered the multicriteria s-t minimum
cut problem, in which the cut must separate two specified vertices, s and t. They proved that
this problem is strongly NP-complete, even for just two criteria.
We show here that (the AND-version of) the multicriteria global minimum cut decision problem can be solved in polynomial time for any fixed number of criteria, making it strictly easier
than its s-t variant. The running time of our algorithm is O(mn2k ), where m = |E| is the number
of edges in the graph, n = |V | is the number of vertices, and k is the number of criteria. This
easily implies tractability of the optimization problem and also yields a pseudo-polynomial algorithm for constructing the Pareto set. The problem, however, becomes strongly NP-hard when
the number of criteria is not fixed. We also show that the directed version of the problem is
strongly NP-hard even for just two criteria.
2.1 Introduction
11
The single-criterion minimum cut problem has been studied for more than four decades as a
fundamental graph optimization problem (see, e.g., [GH94, NI92, KS96, SW97, Kar99, Kar00]).
Minimum cuts are used in solving a large variety of problems, including VLSI design, network
design and reliability, clustering, and more (see [Kar00] and references therein). The best known
deterministic algorithms for this problem run in O(mn + n2 log n) time (Nagamochi and Ibaraki
[NI92] and Stoer and Wagner [SW97]). The best known randomized algorithm runs in O(m log3 n)
time (Karger [Kar00]). (As can be seen, there is a huge gap between the complexities of the
deterministic and randomized algorithms!). These algorithms are faster than the best known
algorithms for the s-t minimum cut problem which are based on network flow.
The polynomial time algorithm for the multicriteria problem relies on the fact that the standard single criterion global minimum cut problem has only a polynomial number of almost optimal
solutions. More specifically, Karger and Stein [KS96] showed that for every α ≥ 1, not necessarily
integral, the number of α-approximate solutions is only O(n2α ). Karger [Kar00] improved this
bound to O(nb2αc ). Nagamochi et al. [NNI97] gave a deterministic O(m2 n + mn2α ) time algorithm for finding all the α-approximate cuts. Our algorithms for the multicriteria problem use
their algorithm.
Apart from the theoretical interest in minimum cuts in the multicriteria setting, there are some
applications in which this problem is of interest. (The multicriteria global minimum cut problem
is of interest in almost any application of the single criterion global minimum cut problem.) A
multicriteria minimum balanced-partition is required, for example, in the situations described
in [SKK99].
A special case of the bicriteria global minimum cut problem, called the ≤ r-cardinality min-cut,
was considered by Bruglieri et al. [BEH00, BME04]. The input to this problem is an undirected
graph G = (V, E) with a single weight function w : E → R+ defined on its edges. The goal is
to find a cut of minimum cost that contains at most r edges. This is exactly the optimization
version of the bicriteria minimum cut problem, where w1 (e) = 1, w2 (e) = w(e), for every e ∈ E,
and w1 (C) must not exceed r. Bruglieri et al. [BEH00, BME04] ask whether this problem can
be solved in polynomial time. We answer their question in the affirmative. We also obtain a
polynomial time algorithm for finding a minimum cut which contains at most r vertices on the
smallest side of the cut.
As mentioned, most research on multicriteria optimization focused, in our terminology, on
AND-versions of various multicriteria optimization problems. We consider here also the ORversions of the global minimum cut problem, the shortest path problem, and the minimum spanning tree problem.
OR-versions of multicriteria optimization problems may be seen as generalizations of the
12
scheduling problem on unrelated machines (see [HS76, LST90, JP99]). The input to such a
scheduling problem is a set of n jobs that should be scheduled on m machines. The i-th job has a
cost vector (ci1 , . . . , cim ) associated with it, where cij is the processing time of the i-th job on the
j-th machine. The goal is to allocate the jobs to the machines so as to minimize the makespan,
i.e., the completion time of the last job. (Jobs allocated to the same machine are processed
sequentially.) This is precisely the OR-version of the min-max m-criteria minimum cut problem
on a graph with two vertices and n parallel edges.
As another example where the OR-version of the multicriteria minimum-cut problem is of
interest, consider a cyber-attacker wishing to disconnect a computer network, where there is more
than one option for damaging each link. For example, assume that each link can either be disconnected by an electronic attack, which requires a certain amount of work hours (that may differ
for different links), or by physically disconnecting it, e.g., by creating a strong electromagnetic
field near the underground cable (the required power may again differ from link to link). Assuming an upper bound on the available power for electromagneric fields, what is the minimum
electronic-attack time which enables disconnecting the network?
It follows immediately from the simple reduction given above that the OR-version of the
multicriteria global minimum cut problem is NP-hard even for just two criteria. We show, however,
that the problem can be solved in pseudo-polynomial time for any fixed number of criteria. We
also show that the problem can be solved in polynomial time when k, the number of criteria,
is fixed and at least k − 1 of the weight functions assume only a fixed number of values. We
also obtain some results on the complexity of the OR-versions of the shortest path and minimum
spanning tree problems.
The rest of this chapter is organized as follows. In the next section we consider the ANDversion of the global minimum cut problem. In Section 2.3 we then consider the OR-version of
the problem. In Section 2.4 we consider the OR-version of the multicriteria shortest path and
minimum spanning tree problems. Finally, we conclude in Section 2.5 with some concluding
remarks and open problems.
2.2
Multicriteria global minimum cut: the AND-version
We first present a polynomial time algorithm for the min-max version of the multicriteria global
minimum cut. The algorithm for solving the min-max version of the problem is then used to solve
the decision and optimization problems.
2.2 Multicriteria global minimum cut: the AND-version
13
Algorithm Min-Max(G(V, E, w1 , ..., wk )):
P
1. Let w0 (e) = ki=1 wi (e), for every e ∈ E.
2. Find all the k-approximate minimum cuts in G with
respect to w0 .
3. Among all the cuts C found in the previous step
find the one for which maxki=1 wi (C) is minimized.
Figure 2.2: A strongly polynomial time algorithm for the min-max version of the k-criteria
global minimum cut problem.
2.2.1
The Min-Max problem
An optimal min-max cut is a cut C for which maxki=1 wi (C) is minimized. We show that the
simple algorithm given in Figure 2.2 solves the min-max version of the k-criteria global minimum
cut problem in polynomial time, for every fixed k. A k-approximate cut in a graph G with respect
to a single weight function w0 is a cut whose weight is at most k times the weight of the minimum
cut.
Theorem 2.1. Algorithm Min-Max solves the min-max version of the k-criteria global minimum
cut problem. For any fixed k, it can be implemented to run, deterministically, in O(mn2k ) time.
Proof. We begin by proving the correctness of the algorithm. We show that if C is an optimal minP
max cut, and D is any other cut in the graph, then w0 (C) ≤ k·w0 (D), where w0 (e) = ki=1 wi (e),
for every e ∈ E. This follows as
w0 (C) =
k
X
i=1
k
k
i=1
i=1
wi (C) ≤ k · max wi (C) ≤ k · max wi (D) ≤ k ·
k
X
wi (D) = k · w0 (D) .
i=1
The inequality k · maxki=1 wi (C) ≤ k · maxki=1 wi (D) follows from the assumption that C is an
optimal min-max cut. In particular, if D is an optimal minimum cut with respect to the single
weight function w0 , then w0 (C) ≤ k·w0 (D), and it follows that C is a k-approximate cut of G with
respect to w0 . This proves the correctness of the algorithm.
We next consider the complexity of the algorithm. Karger and Stein [KS96] showed that every
graph has at most O(n2k ) k-approximate cuts and gave a randomized algorithm for finding an
implicit representation of them all in Õ(n2k ) time. A deterministic algorithm of Nagamochi et al.
14
[NNI97] explicitly finds all the k-approximte cuts in O(mn2k ) time. Choosing the best min-max
cut among all the k-approximate cuts also takes only O(mn2k ) time.
It is also easy to see that for any 1 < α ≤ k, we can find an α-approximate solution to the
min-max problem in O(mn2k/α ) time, by checking all the k/α-approximate cuts in G0 = (V, E, w0 ).
The randomized algorithm of Karger and Stein [KS96] extends to finding all the k-approximate
minimum r-cuts in Õ(n2k(r−1) ) time. (An r-cut is a partition of the graph vertices into r sets,
instead of 2). Thus, it is easy to see that the min-max multicriteria problem can also be solved
for r-cuts, in Õ(mn2k(r−1) ) time using this randomized (Monte-Carlo) algorithm.
2.2.2
The decision problem
We next show that the algorithm for the min-max version of the k-criteria problem can be used
to solve the decision version of the problem: Given k bounds b1 , b2 , . . . , bk , is there a cut C such
that wi (C) ≤ bi , for 1 ≤ i ≤ k.
Theorem 2.2. For any fixed k, the decision version of the k-criteria global minimum cut problem
can be solved, deterministically, in O(mn2k ) time.
Proof. The decision problem can be easily reduced to the min-max problem. Given k weight
functions w1 , . . . , wk : E → R+ and k bounds b1 , b2 , . . . , bk , we simply produce scaled versions
wi0 (e) = wi (e)/bi , for every e ∈ E and 1 ≤ i ≤ k, of the weight functions. Clearly the answer to
the decision problem is ‘yes’ if and only if there is a cut C for which maxki=1 wi0 (C) ≤ 1.
2.2.3
The optimization problem
We next tackle the optimization problem: Given k − 1 bounds b1 , b2 , . . . , bk−1 , find a cut C for
which wk (C) is minimized, subject to to the constraints wi (C) ≤ bi , for 1 ≤ i ≤ k − 1.
Theorem 2.3. For any fixed k, the optimization version of the k-criteria global minimum cut
problem for graphs with integer edge weights can be solved, deterministically, in O(mn2k log M )
P
time, where M = e∈E wk (e).
Proof. If the k-th weight function assumes only integral values, we can easily use binary search
to solve the optimization problem. Given the k − 1 bounds b1 , b2 , . . . , bk−1 , we conduct a binary
search for the minimal value bk for which there is a cut C such that wi (C) ≤ bi , for 1 ≤ i ≤ k.
As the minimal bk is an integer in the range [0, M ], this requires the solution of only O(log M )
decision problems.
15
The algorithm given above is not completely satisfactory as it is not strongly polynomial and
does not work with non-integral weights. These problems can be fixed, however, as we show
below.
Theorem 2.4. For any fixed k, the optimization version of the k-criteria global minimum cut
problem for graphs with arbitrary real edge weights can be solved, deterministically, in O(mn2k log n)
time.
Proof. We first isolate a small interval that contains the minimal value of bk . Let S = {wk (e) | e ∈
E} be the set of values assumed by the k-th weight function. The minimum bk of the optimization
problem lies in an interval [s, ms], for some s ∈ S. (If C is the cut that attains the optimum, let s
be the weight of the heaviest edge, with respect to wk , in the cut.) Using a binary search on the
values in S, we can find such an interval that contains the minimum. This requires the solution
of only O(log m) = O(log n) decision problems. Next, we conduct a binary search in the interval
[s, ms] until we narrow it down to an interval of the form [s0 , (1 + n1 )s0 ] which is guaranteed to
contain the right answer. This again requires the solution of only O(log(mn)) = O(log n) decision
problems.
Next, we run a modified version of the Min-Max algorithm given in Figure 2.2 on the following
scaled versions of the weights: wi0 (e) = wi (e)/bi , for 1 ≤ i ≤ k − 1, and wk0 (e) = wk (e)/s0 . It
is easy to see that if C is an optimal solution of the optimization problem, then C is also an
(1 + n1 )-approximate solution of the min-max problem. This in turn implies, as in the proof of
Theorem 2.1, that C is also a k(1 + n1 )-approximate minimum cut with respect to the weight
P
function w0 (e) = ki=1 wi (e), for every e ∈ E. Instead of finding all the k-approximate minimum
cuts with respect to w0 , as done by algorithm Min-Max, we find all the k(1 + n1 )-approximate
minimum cuts. Among all these cuts we find a cut C for which wi0 (C) ≤ 1, i.e., wi (C) ≤ bi ,
for 1 ≤ i ≤ k − 1, and for which wk (C) is minimized. This cut is the optimal solution to the
optimization problem.
We next analyze the complexity of the algorithm.
be solved in
O(mn
n1/n
1
2k(1+ n
)
O(mn2k
) =
log n) time.
O(mn2k )
All the k(1 +
The O(log n) decision problems can
1
n )-approximate
cuts can then be found in
time using the algorithm of Nagamochi et al. [NNI97]. (Note that
= O(1).) Checking all these cuts also takes only O(mn2k ) time. This completes the proof of
the theorem.
2.2.4
Two applications
Theorem 2.5. Let G = (V, E) be an undirected graph and let w : E → R+ be a weight function
defined on its edges. Let 1 ≤ r ≤ m. Then, there is a deterministic O(mn4 log n) time algorithm
16
for finding a cut of minimum weight that contains at most r edges.
Proof. We simply let w1 (e) = 1 and w2 (e) = w(e), for every e ∈ E, and solve the optimization
problem with b1 = r.
As mentioned in the introduction, this solves an open problem raised by Bruglieri et al.
[BEH00, BME04]. We also have:
Theorem 2.6. Let G = (V, E) be an undirected graph and let w : E → R+ be a weight function
defined on its edges. Let 1 ≤ r ≤ n. Then, there is a deterministic O(n6 log n) time algorithm for
finding a cut of minimum weight with at most r vertices on its smaller side.
Proof. We set up two weight functions over a complete graph on n = |V | vertices: w1 (u, v) = 1,
for every u, v ∈ V , and w2 (u, v) = w(u, v), if (u, v) ∈ E, and w(u, v) = 0, otherwise. We then find
a cut C that minimizes w2 (C) subject to the constraint w1 (C) ≤ r(n − r).
2.2.5
The Pareto Set
Suppose all weight functions are integral. Let Mi =
P
e∈E
wi (e), for 1 ≤ i ≤ k. The Pareto set
can be trivially found by invoking the basic decision algorithm Πki=1 Mi times, or the optimization
algorithm Πk−1
i=1 Mi times. These naive algorithms are pseudo-polynomial for every fixed k.
Using very similar ideas we can also obtain an FPTAS for finding an approximate Pareto
set, a notion defined by Papadimitriou and Yannakakis [PY00]. It is defined as a set of feasible
k-tuples, such that for every solution there is a k-tuple in the set within a factor of (1 − ) in all
coordinates. More formally, the set P (G, w1 , . . . , wk ) is a set of cost vectors of cuts in the graph
such that for every cut C there exists (c1 , ..., ck ) ∈ P (G, w1 , .., wk ) such that (1 − )ci ≤ wi (C)
for 1 ≤ i ≤ k. It is easy to see that we can find this set in polynomial time, by invoking the basic
algorithm for the decision problem only for powers of (1 − ), instead of checking all the possible
values.
2.2.6
Hardness results
Theorem 2.7. The multicriteria minimum cut problem with a non-fixed number of criteria is
strongly NP-complete.
Proof. We use a reduction from the bisection width problem (see [GJ79], problem ND17): Given
an unweighted input graph G = (V, E) on n = 2r vertices and a bound b, is there a bisection
17
Figure 2.3: A simple illustarion (with n = 4) for the construction in the proof of Theorem
2.7. All the original edges of the graph (solid lines) are assigned the costs (0, 0, 0, 0, 0, 0, 1).
The edges connected to s are assigned the costs (1, 1, 1, 1, 1, 0, 0), and the edges connected
to t are assigned the costs (1, 1, 1, 1, 0, 1, 0).
of the graph that cuts at most b edges? We transform such an instance in the following way:
Assume that V = {1, 2, . . . , n}. We add two vertices, s and t, and add edges connecting them
to each of the vertices in V . Let G0 = (V 0 , E 0 ) be the resulting graph. Each edge of G0 is now
assigned n + 3 weights. For 1 ≤ i ≤ n, we let wi (s, i) = wi (t, i) = 1, and wi (e) = 0 for all other
edges. We assign wn+1 (e) = 1 for the edges of the form (s, i), i ∈ V , and wn+1 (e) = 0, otherwise.
Similarly, we assign wn+2 (e) = 1 for the edges of the form (t, i), i ∈ V , and wn+2 (e) = 0, otherwise.
Finally, wn+3 (e) = 1 for e ∈ E, and wn+3 (e) = 0, otherwise. It is now easy to see that G has a
bisection of width at most b if and only if G0 has a cut C for which wi (C) ≤ 1, for 1 ≤ i ≤ n,
wn+1 (C), wn+2 (C) ≤ r, and wn+3 (C) ≤ b.
It is also not difficult to show that the directed multicriteria global minimum cut problem
is strongly NP-complete, even for two criteria. In this problem, we are given a directed graph
G = (V, E) with weight functions w1 , ..., wk : E → R+ . A solution consists of a cut C, and of a
labelling of the two vertex sets it separates by S and T . The weights of each cut are the sums
of the weights of the cut edges directed from S to T . Each of the above mentioned variants for
the undirected multicriteria minimum cut problem can be considered here as well: the decision,
optimization, min-max and Pareto-set problems. In the directed multicriteria s-t min-cut problem,
two vertices, s and t, are specified with the input, and the solution must satisfy: s ∈ S and t ∈ T .
Theorem 2.8. The directed multicriteria global minimum cut problem is strongly NP-complete,
even for just two criteria.
18
Proof. We show this by a reduction from undirected multicriteria s-t-min-cut, which is strongly
NP-hard, even for k = 2 (Papadimitriou and Yannakakis [PY00]).
An instance of the undirected bicriteria s-t-min-cut decision problem can be reduced to an
instance of the directed bicriteria s-t-min-cut problem simply by replacing each edge by two antiparallel directed edges with the same weight. Recall that in a directed s-t-min-cut only edges
directed from S to T contribute to the cut weights (s ∈ S and t ∈ T ), so having edges in the
opposite direction does not influence the solution.
This instance can then be reduced to an instance of the directed bicriteria global min-cut
decision problem. We simply connect each vertex to s with edges having weight m · M + 1 in
both criteria (where M is the maximal weight), and do the same from t to all the other vertices
(if some of these edges already exist then we replace them). We assume that at least one input
bound satisfies bi < m · M + 1, otherwise the answer is trivially ”yes”. So a solution to this
problem will necessarily have s and t on different sides, s ∈ S and t ∈ T . Therefore a solution
to this problem also solves the original problem, and it has the same weights. Thus, the directed
multicriteria global min-cut decision problem is strongly NP-hard, even for k = 2.
2.3
Multicriteria global minimum cut: the OR-version
2.3.1
Relation to scheduling on unrelated machines
As mentioned in the introduction, there is a trivial reduction from the scheduling on unrelated
machines problem to the OR-version of the min-max multicriteria global minimum cut problem.
Known hardness results for the scheduling problem (see Lenstra et al. [LST90]) then imply the
following:
Theorem 2.9. The OR-version of the min-max multicriteria global minimum cut problem is
NP-hard even for just two criteria. The problem with a non-fixed number of criteria cannot be
approximated to within a ratio better than 3/2, unless P=NP.
The scheduling problem on a fixed number of unrelated machines can however be solved in
pseudo-polynomial time. Horowitz and Sahni [HS76] present a simple branch-and-bound pseudopolynomial algorithm for that problem which runs in O(m2 (kM )k−1 ) time, where m is the number
of jobs, k is the number of machines, and M is the optimal makespan. This immediately implies:
Theorem 2.10. Let G = (V, E) be an undirected graph with k integral weight functions w1 , . . . , wk :
E → N defined on it edges. Let C be a cut in G. Then, a choice function α : E(C) → {1, 2, . . . , k}
which minimizes maxki=1 wi (C, α) can be found in pseudo-polynomial time.
2.3 Multicriteria global minimum cut: the OR-version
19
Algorithm Min-Max-Or(G(V, E, w1 , ..., wk )):
1. Let w0 (e) = minki=1 wi (e), for every e ∈ E.
2. Find all the k-approximate minimum cuts in G with
respect to w0 .
3. For each of the cuts C found in the previous
step, find the best choice function α : E(C) →
{1, 2, . . . , k}.
4. Output the best cut and choice function found.
Figure 2.4: A pseudo-polynomial time algorithm for the min-max version of the k-criteria
global minimum cut problem.
Jansen and Porkolab [JP99] obtained an FPTAS for the unrelated machines scheduling problem, which runs in O(m(k/)O(k) ) time (for any fixed number k of machines). It can be used
instead of the exact algorithm of [HS76] when approximate solutions are acceptable.
2.3.2
The min-max version
We show that the simple algorithm given in Figure 2.4, which is a variant of the algorithm given
in Figure 2.2, solves the OR-version of the min-max problem in pseudo-polynomial time, for any
fixed number of criteria.
Theorem 2.11. The OR-version of the min-max k-criteria global minimum cut problem with
integer edge weights can be solved in O(m2 n2k (kM )k−1 ) time, where M is the optimal min-max
value.
Proof. We begin again with the correctness proof. Let C be an optimal min-max cut and let α
be the corresponding optimal choice function. Let D be any other cut. We show that w0 (C) ≤
k·w0 (D), where w0 (e) = minki=1 wi (e), for every e ∈ E. To see that, we let β : E(D) → {1, 2, . . . , k}
be a choice function for which β(e) = i if wi (e) ≤ wj (e), for every 1 ≤ j ≤ k. Then,
k
k
i=1
i=1
w0 (C) ≤ k· max wi (C, α) ≤ k· max wi (D, β) ≤ k·w0 (D) .
The second inequality follows as (C, α) is an optimal solution of the min-max problem.
20
We next consider the complexity of the algorithm. The k-approximate cuts with respect to w0
can be found again in O(mn2k ) time using the algorithm of Nagamochi et al. [NNI97]. For each one
of the O(n2k ) approximate cuts produced, we find an optimal choice function using the algorithm
of Horowitz and Sahni [HS76]. The total running time is then O(mn2k + n2k · m2 (kM )k−1 ) =
O(mn2k + m2 n2k (kM )k−1 ), where M is the value of the optimal solution.
Theorem 2.12. The OR-version of the min-max k-criteria global minimum cut problem, with k
fixed, admits an FPTAS.
Proof. The proof is identical to the proof of Theorem 2.11 with the exact algorithm of Horowitz
and Sahni [HS76] replaced by the FPTAS of Jansen and Porkolab [JP99].
As in Section 2.2, we can use the algorithm for the min-max version of the problem to solve
the decision and optimization versions of the problem. We omit the obvious details.
2.3.3
A case that can be solved in polynomial time
We now discuss a restriction of the min-max problem that can be solved in strongly polynomial
time. For simplicity, we consider the bicriteria problem.
Theorem 2.13. Instances of the OR-version of the min-max bicriteria global minimum cut
problem in which one of the weight functions assumes only r different values can be solved in
O(mr+1 n4 ) time.
Proof. Assume, without loss of generality, that w2 assumes only r different real values a1 , a2 , . . . , ar .
Let Ei = w2−1 (ai ) = {e ∈ E | w2 (e) = ai }, for 1 ≤ i ≤ r. Consider an optimal min-max cut C
and an optimal choice function α : E(C) → {1, 2} for it. It is easy to see that for every 1 ≤ i ≤ r
there is a threshold ti such that if e ∈ Ei , then α(e) = 1 if and only if w1 (e) ≤ ti . (Indeed, if there
are two edges e1 , e2 ∈ Ei such that w1 (e1 ) < w1 (e2 ), α(e1 ) = 2 and α(e2 ) = 1, then the choice
function α0 which reverses the choices of α on e1 and e2 is a better choice function. We assume
here, for simplicity, that all weights are distinct.) As there are at most m + 1 essentially different
thresholds for each set Ei , the total number of choice functions that should be considered is only
O(mr ). With a given choice function α : E → {1, 2}, the problem reduces to an AND-version of
the problem with the weights wi0 (e) = wi (e), if α(e) = i, and wi0 (e) = 0, otherwise, for i = 1, 2.
As each such problem can be solved in O(mn4 ) time, the total running time of the resulting
algorithm is O(mr+1 n4 ).
2.4 OR-versions of other multicriteria problems
2.4
21
OR-versions of other multicriteria problems
In this section we consider the OR-versions of the bicriteria shortest path and minimum spanning
tree problems. Our results can probably be extended to any fixed number of criteria.
2.4.1
Shortest paths
The input to the problem is a directed graph G = (V, E) with two weight functions w1 , w2 :
E → R+ defined on its edges, two vertices s, t ∈ V , and two bounds b1 , b2 . The question is
whether there is a path P from s to t in the graph and a choice function α : P → {1, 2} such
that w1 (α−1 (1)) ≤ b1 and w2 (α−1 (2)) ≤ b2 . The graph G = (V, E) may, for example, represent
the map of a city. Each edge e ∈ E of the graph can be traversed either by bus or by subway.
The weight w1 (e) is the number of bus tokens needed for traversing the edge e by bus, while the
weight w2 (e) is the number of subway tokens needed to traverse e by subway. The question then
is whether it is possible to get from s to t using given amounts of subway tokens and bus tokens.
It is easy to see, using a simple reduction from the scheduling on unrelated machines problem,
that the OR-version of the bicriteria shortest path problem is NP-hard. We show, however, that
it can be solved in pseudo-polynomial time. An FPTAS for the problem is easily obtained by
scaling.
Theorem 2.14. The OR-version of the bicriteria shortest path decision problem with integer edge
lengths can be solved in O(nmW log(nW )) time, where W = maxe∈E w1 (e).
Proof. The OR-version of the problem can be easily reduced to the AND-version of the problem
by replacing each edge e having a weight vector (w1 (e), w2 (e)) by two parallel edges e0 and e00
having weight vectors (w1 (e), 0) and (0, w2 (e)). The standard, AND-version, of the problem can
be solved using an algorithm of Hansen [Han80] within the claimed time bound.
2.4.2
Minimum spanning trees
Next we consider the OR-version of the bicriteria minimum spanning tree problem. The input is
an undirected graph G = (V, E), two weight functions w1 , w2 : E → R, and two bounds b1 and
b2 . The question is whether there exist a spanning tree T and a choice function α : T → {1, 2}
such that w1 (α−1 (1)) ≤ b1 and w2 (α−1 (2)) ≤ b2 .
The OR-version of the bicriteria minimum spanning tree problem is again easily seen to be
NP-hard. We provide a polynomial time algorithm for a special case of the problem, and a
pseudo-polynomial time algorithm for the general case.
22
Theorem 2.15. The OR-version of the minimum spanning tree problem in which one of the
weight functions is constant, i.e., w2 (e) = c, for every e ∈ E, can be solved by solving a single
standard minimum spanning tree problem.
Proof. We simply solve the standard minimum spanning tree problem with respect to the weight
function w1 and obtain a minimum spanning tree T . For the bb2 /cc heaviest edges of T we choose
to pay the w2 cost, and for all the others we pay the w1 cost. The correctness of this procedure
follows from the well known fact that if the weights of the edges of T are a1 ≤ a2 ≤ · · · ≤ an−1 ,
and if T 0 is any other spanning tree of the graph G with edge weights a01 ≤ a02 ≤ · · · ≤ a0n−1 , then
ai ≤ a0i , for 1 ≤ i ≤ n − 1.
Theorem 2.16. The OR-version of the bicriteria minimum spanning tree decision problem with
integer edge lengths can be solved in O(n4 b1 b2 log(b1 b2 )) time.
Proof. The OR-version of the problem can be easily reduced to the AND-version of the problem
by replacing each edge e having a weight vector (w1 (e), w2 (e)) by two parallel edges e0 and e00
having weight vectors (w1 (e), 0) and (0, w2 (e)). The standard, AND-version, of the problem can
be solved using an algorithm of Hong et al. [HCP04] within the claimed time bound.
The AND-version of the multicriteria minimum spanning tree problem can be solved in polynomial time using matroid intersection algorithms.
2.5
Concluding remarks
We showed that the standard (i.e., the AND-version) multicriteria global minimum cut problem
can be solved in polynomial time for any fixed number k of criteria. The running time of our
algorithm, which is O(mn2k ), is fairly high, even for a small number of criteria. Improving this
running time is an interesting open problem. We also considered the OR-version of the problem
and showed that it is NP-hard even for just two criteria. It can be solved, however, in pseudopolynomial time, and it also admits an FPTAS, for any fixed number of criteria. Finally, we
considered the OR-versions of the bicriteria shortest path and minimum spanning tree problems,
and showed that both of them are NP-hard but can be solved in pseudo-polynomial time. It will
also be interesting to study OR-versions of other multicriteria optimization problems.
Chapter 3
Cooperative TSP
3.1
Introduction
The Traveling Salesperson Problem (TSP) is a classical problem in combinatorial optimization,
which has been studied extensively in many forms. Cooperative TSP is a set of variants of TSP
in which the customers are able to move in order to assist the selling process. They may move in
order to expedite the deliveries, and may also move after meeting the salesperson in order to help
the distribution of the goods. The basic motivation for these variants is that the “salesperson” and
“customers” are often part of the same organization, and can be instructed by the “headquarters”
to cooperate. For example, consider a secret message that has to be distributed to several spies,
but is only allowed to be passed in person. Every spy can be instructed when and where to receive
it, and a spy who receives the message may then assist by passing it forward. We may want to
devise a scheme for delivering the secret to all the spies as fast as possible. A further illustration
is the problem of sleeping robots which need to be woken up. Once a robot is awaken, it can be
instructed to assist in waking-up others, but it cannot move before that. As a robot’s battery is
limited, we may wish to minimize the maximal travel of any of them. This problem is related to
the previously studied “Freeze-Tag” problem [ABF+ 02, SABM02, ABG+ 03, KLS04] (which we
later describe in more detail).
Formally, an instance of Cooperative TSP (cTSP) is a set of agents and a salesperson, located
in a finite metric space or a Euclidean space. A solution is a synchronized series of move instructions to all participants (i.e., the salesperson and the agents), such that all the agents eventually
receive the delivery. We next elaborate on the various cooperation modes, the cost of solutions
and other parameters affecting the cTSP.
24
Cooperative TSP
Cooperation Modes. We consider three modes of cooperation. In the Purchase-Cooperation mode the salesperson has to meet all agents, and the agents are allowed to move towards the
salesperson. In the Sales-Cooperation mode, each agent receiving a delivery becomes capable
of making deliveries (exactly like the salesperson). However, an agent is not allowed to move
before receiving a delivery. In the Full-Cooperation mode, an agent may cooperate in both
the purchase and sales phases. That is, an agent may move before receiving the delivery and may
make deliveries after receiving it.
Goal Functions. Three objectives are considered for Cooperative TSP: Minimizing the total
travel of all the participants (Min-Sum), minimizing the maximal travel of any participant (MinMax), and minimizing the total time until the sales process ends (Min-Makespan). Naturally,
the Min-Sum goal is motivated by scenarios in which the travel of all the participants is covered
by the same entity (e.g., the delivery-service company), which is therefore interested in minimizing
the total travel. The Min-Max objective is required, for example, when there is a bound on the
amount of fuel/battery of each participant, and each of them should spend as little energy as
possible on this delivery process. Min-Makespan is motivated by cases in which the completion
of the deliveries is urgent.
Metric Space. We consider Cooperative TSP in any fixed-dimension Euclidean space and in
non-negative weighted undirected graphs. Note that w.l.o.g., we may assume that the graph is
complete and that the weights of all edges satisfy the triangle inequality, hence this is a finite
metric space.
Roundtrip vs. Path. We consider both roundtrip versions, in which all participants are
required to return to their initial location, and path versions in which there is no such requirement
(the starting points may be arbitrary, or we may only be interested in what happens until the
deliveries are made).
In this study we consider all the problems arising from combining cooperation-modes, goal
functions, graph/Euclidean space and path/roundtrip versions. We refer to each of the problems we study using the format: Goal-Function - Cooperation-Mode - [Euclidean] - cTSP
(e.g., “Min-Sum Purchase Euclidean cTSP”). Unless explicitly stated otherwise, a problem
name indicates its path version (rather than its roundtrip version).
3.1 Introduction
25
cTSP in Graphs
Goal
Min
Sum
Min
path
Max
Purchase
Cooperation
Approx. Inapprox.
2 + ln 3
NP-hard
PTAS
no FPTAS
Min
Polynomial
Makespan
Min
Sum
Min
round
Max
Min
trip
Makespan
3
2
131
130
PTAS
−ε
no FPTAS
Polynomial
Sales
Cooperation
Approx.
Inapprox.
2
3
√
O( log n)
∗
Full
Cooperation
Approx. Inapprox.
NP-hard
2 + ln 3
APX-hard
2−ε
4
2−ε
∗∗
2
2−ε
−ε
3
2
5
3
−ε
3
2
131
130
131
130
3
√
O( log n)
3
2
−ε
2
2−ε
5
4
−ε
2
2−ε
−ε
Table 3.1: Approximation ratios vs. inapproximability ratios for cTSP in weighted Graphs.
(∗) is by [KLS04] and (∗∗) is by [ABF+ 02]. The parameter ε stands for an arbitrarily small
positive constant, or for a positive function that tends to zero as the input size increases.
3.1.1
Related Studies
The classical TSP problem remains NP-hard even in planar graphs [GGJ76, Pap77]. However,
there is a PTAS for any fixed-dimension Euclidean space [Aro98, Mit99]. When only metric space
is assumed, the best known approximation algorithm yields a 32 -approximation ratio [Chr76] and
an inapproximability factor of
131
130 -
was shown [EK01].
The Freeze-Tag Problem. The Freeze-Tag problem was first suggested and studied by
Arkin et al. in [ABF+ 02]. The problem arises in the context of swarm robotics: How to wake up
a set of slumbering robots, when initially only one robot is awake (waking up a robot requires
reaching its location). Once a robot is woken up it can assist in waking up other slumbering
robots. The objective is to have all robots awake as early as possible. In our terminology, this
is the path version of Min-Makespan Sales cTSP. Arkin et al. [ABF+ 02] provided an NPhardness proof, a PTAS for the Euclidean variant, and a constant approximation for some graph
families. A series of studies followed (e.g., [SABM02, ABG+ 03, KLS04]) culminating with an
√
O( log n)-approximation for the general weighted graph case [KLS04].
26
Cooperative TSP
TSP with Neighborhoods. TSP with Neighborhoods is a proximity-related variant of TSP.
In this problem each customer is willing to meet the salesperson anywhere within some neighborhood. The problem was first studied by Arkin et al. [AH94], followed by quite a few papers (e.g.,
[MM95, GL99, DM01, dBGK+ 05, SS05, Mit07]). An instance of TSP with Neighborhoods may
reside in a weighted graph or in a Euclidean space. The problem seems quite related to Purchase cTSP, as in both customers are willing to approach the salesperson. However, in TSP with
Neighborhoods the customers’ travel is not counted in the goal function, while in Cooperative
TSP their moves do cost, and are part of the optimization task.
Other Cooperative Multi-Agent Routing Problems. As noted in [ABF+ 02], the
Freeze-Tag Problem (and thus the Cooperative TSP problems) bears features of broadcasting,
routing, scheduling and network design. The minimum broadcast time, the multicast problem and
the minimum gossip time problem are all closely related to Cooperative TSP (see [HHL88] for
a survey and [Rav94, BNGNS98] for approximation results). Controlling swarms of robots in
order to perform a certain task, has also been studied in various algorithmic aspects, including
environment exploration, robot formation, searching and recruitment (see [ABF+ 02] for a list of
relevant papers). Other researches confront similar scenarios, but with no central control, where
each agent has to make decisions with limited knowledge regarding the environment and the other
agents (for example, the problem of routing autonomous agents in a wireless sensor network, and
ants behavior inspired algorithms; see [ABF+ 02] for a list of relevant papers).
As cTSP generalizes the Freeze-Tag problem and is closely related to the TSP with Neighborhoods problem, the algorithms (and intractability results) obtained for cTSP apply to similar
scenarios, e.g., cooperative robots tasks (for additional relevant scenarios see [AH94, ABF+ 02]).
3.1.2
Our Contribution
We consider all combinations of cooperation modes, goal functions, path / roundtrip and graph
/ Euclidean versions. The results for cTSP in weighted graphs are summarized in Table 3.1
and the results for cTSP in a fixed-dimension Euclidean space are summarized in Table 3.2.
We obtain constant approximations for most of the problems, PTAS for many of them, and
polynomial-time exact solutions for a few. On the intractability side we obtain NP-hardness
proofs and inapproximability factors for all the NP-hard graph problems and for some of the
Euclidean problems.
3.2 Euclidean cTSP
27
Euclidean cTSP
path
round
trip
Goal
Min-Sum
Min-Max
Min-Makespan
Min-Sum
Min-Max
Min-Makespan
Purchase
Sales
Full
Cooperation Cooperation Cooperation
5
+ε
PTAS
2+ε
3
PTAS
3
4
∗
Polynomial
PTAS
PTAS
PTAS
PTAS
PTAS
PTAS
3
2
Polynomial
PTAS
PTAS
Table 3.2: Approximation ratios for cTSP in any fixed dimension Euclidean space. (∗) is
by [ABF+ 02]. The parameter ε stands for an arbitrarily small positive constant, or for a
positive function that tends to zero as the input size increases.
Chapter Organization. We start with our results for Euclidean cTSP, which we present
in Section 3.2. The results for cTSP in weighted graphs are presented in Section 3.3. Each section
is divided into three subsections, one for each of the goal functions (Min-Sum, Min-Max and
Min-Makespan, in this order). Furthermore, each subsection is divided according to cooperation
modes (Purchase, Sales and Full-Cooperation, in this order).
3.2
Euclidean cTSP
This section presents the results we obtained for the various Euclidean cTSP problems.
3.2.1
Min-Sum Euclidean-cTSP
In this subsection we consider the various objectives for the path versions of Min-Sum Euclidean-cTSP. It is not hard to see that all the roundtrip versions, with either of the three
cooperation-modes, are identical to the classical TSP problem. Specifically,
Claim 3.1. For any metric space M, the roundtrip versions of Min-Sum Purchase cTSP
in M, Min-Sum Sales cTSP in M, and Min-Sum Full-Cooperation cTSP in M are all
equivalent to TSP in M.
Proof. Consider a solution for any of the above cTSP problems in M. In addition, consider the
28
Cooperative TSP
first meeting between the salesperson and an agent who moves in this solution. Let x be the point
in which this meeting occurs, and let y be the initial location of that agent.
We observe that since each agent’s moves form a cycle, there is a solution with the same cost
in which that agent does not move. This holds since the salesperson can travel from x to y along
the path traveled by the agent, meet the agent at x, then follow the rest of the cycle traveled by
that agent (in reverse order), and return back to y. Thus, exactly the same points are visited and
the cost of travel remains the same.
Therefore, w.l.o.g., in any solution of the above mentioned cTSP problems, no participant
moves except the salesperson. Hence, all the above mentioned cTSP problems in M are equivalent
to TSP in M.
Thus, like the classic TSP problem, these roundtrip problems are all NP-hard [GGJ76,
Pap77], and have a PTAS for any fixed-dimension Euclidean space [Aro98, Mit99]. We therefore
consider the Path version of these problems.
3.2.1.1
Min-Sum Purchase Euclidean-cTSP
We next provide a PTAS for Min-Sum Purchase Euclidean-cTSP. Note that the problem is
NP-hard even for the planar case. This follows, since an instance of the classical planar TSP can
be reduced to an instance of Min-Sum Purchase Euclidean-cTSP by simply replacing each
customer with three agents. This makes the salesperson the only participant who moves in an
optimal solution (moving such three agents to meet him at another location clearly costs more
than moving the salesperson from the other location to their initial location and back).
The algorithm and analysis below use Arora’s technique for the PTAS of Euclidean TSP
[Aro98]. Our algorithm differs from Arora’s algorithm in that it has to consider all the agents’
paths and not only the salesperson’s path. We show how this can be done while keeping the
dynamic programming polynomial. We get:
Theorem 3.2. Min-Sum Purchase Euclidean-cTSP admits a PTAS.
We describe the PTAS for the 2-dimensional case. The extension to any fixed dimension is
straightforward. Roughly speaking, we prove the existence of a coarse solution, which is called
a minimal cost portal-limited-solution, that has a cost of at most (1 + ε) the cost of an optimal
solution. We then show how to find a minimal cost portal-limited-solution in polynomial time,
using dynamic programming. We start by introducing the terminology. Readers familiar with
Arora’s PTAS for Euclidean TSP may want to skip to the (slightly altered) definition of portallimited-solutions.
3.2 Euclidean cTSP
29
Let ε > 0 be an arbitrary small constant. Denote by n the number of participants and by
OP T the cost of the optimal solution. Let L = 23+d2 log ne (the smallest power of 2 which is at
least 8n2 ). By stretching and shifting the input points we may assume, without loss of generality,
that all the participants are located inside the bounding box [0, L/2]2 and that OP T > L/4.
Super-pixels. We call each square [j, j + 2] × [j 0 , j 0 + 2], where j, j 0 ∈ {0, 2, 4, . . . , L − 2}, a
pixel. We name the point (j + 1, j 0 + 1) the center of the pixel [j, j + 2] × [j 0 , j 0 + 2]. For every
i = 0, . . . , log L − 1, we call each square [j, j + L/2i ] × [j 0 , j 0 + L/2i ], where j, j 0 ∈ {0, L/2i , 2 ·
L/2i . . . , L − L/2i }, a super-pixel of level i. Thus, each super-pixel of level log L − 1 is a pixel and
the super-pixel of level 0 is the entire bounding box. Additionally, note that different super-pixels
of the same level may overlap only at their boundaries, and that each super-pixel of level i contains
four super-pixels of level i + 1, for i = 0, . . . , log L − 2. Clearly, the total number of super-pixels
is polynomial in n. From now on we consider, without loss of generality, only instances for which
all the participants are located at pixel centers. This is possible since any optimal solution of a
general instance can be modified by instructing each participant to initially move to its pixel’s
√
center. This increases the cost of the solution by at most n · 2. As OP T > L/4 ≥ 2n2 , the
increase is at most OP T /n, which is less than ε/2 · OP T , for a sufficiently large n.
An (a, b)-shifting. Let 0 ≤ a, b < L/2 be two even integers. For a set A ⊆ [0, L/2]2 we define
the (a, b)-shift of A to be the set {(x + a, y + b) | (x, y) ∈ A}. In particular, we are interested in
an (a, b) shift of the original instance, (a, b)-shifted instance, which by our choice of parameters
lies inside the bounding box
[0, L]2 .√
√
Portals. Let m ∈ [ 8
2 log L 16 2 log L
,
)
ε
ε
be a power of 2. Note that, m = O( logε n ). For each
super-pixel we mark each one of its four boundaries with m equidistant points that we refer to
as portals. In particular, the portals include the four corners of the super-pixel. Note that, as m
is a power of 2, each portal of a super-pixel of level i is also a portal of a smaller super-pixel of
level i + 1, for i = 0, . . . , log L − 2. This is illustrated in Figure 3.1.
Portal-limited-solutions. We define a portal-limited-solution as a solution that satisfies the
following four conditions:
1. Each participant may cross the boundary of a super-pixel only at its portals.
2. The salesperson does not cross her own route except at portals, where she may visit at most
twice.
3. A meeting between an agent and the salesperson occurs only at a pixel center.
4. If two (or more) agents happen to reside at a pixel, then they all travel to (or stay at) the
pixel’s center and cease to move.
30
Cooperative TSP
Figure 3.1: This figure illustrates the partition of each super-pixel into four smaller superpixels, and the definition of portals (m on each super-pixel boundary). Note that a portal
of the bigger super-pixel (black) is also a portal of the super-pixels contained in it (grey).
Therefore, in a portal-limited-solution, the tour of each participant is a collection of segments
which connect portals to portals, and centers of pixels to portals. Additionally, in an optimal
portal-limited-solution tours of two agents do not cross.
Using the above notations, our PTAS relies on the following two Lemmata:
Lemma 3.3. A minimal cost portal-limited-solution can be found in time polynomial in n.
Lemma 3.4. Let a, b be two even integers chosen uniformly at random from the set {0, 2, . . . , L/2−
2}. Then, the expected cost of a minimal cost portal-limited-solution of the (a, b)-shifted instance,
is at most (1 + ε) · OP T .
The PTAS enumerates over all O(L2 ) values of (a, b) pairs. For each pair it applies Lemma 3.3
to find a minimal cost portal-limited-solution. Finally, it outputs the cheapest solution found,
which according to Lemma 3.4, must have a cost of at most (1 + ε) · OP T . Clearly, the O(n4 )
factor in running time, caused by the enumeration over all (a, b) pairs, can be avoided if only an
expected (1 + ε) · OP T cost is desired.
The proof of Lemma 3.3 explains how to consider both the salesperson’s and the other agents’
paths, while keeping the time polynomially bounded.
Proof. (of Lemma 3.3) We use dynamic programming to build a polynomial-size table. For
each super-pixel, the table contains 64m = nO(1/ε) entries. For each entry we store portions of
some portal-limited-solutions (the portions of solutions limited to that super-pixel) together with
their contribution to the overall cost.
3.2 Euclidean cTSP
31
The construction of the table is conducted in a bottom-up manner, starting from the pixels.
A minimal value portal-limited-solution for the whole instance is obtained at the bounding-box
super-pixel.
The entries of the table for each super-pixel are represented by a list of 4m elements, one
element for each portal of the super-pixel. Each element takes one of the following six values:
1. The salesperson enters the super-pixel at this portal
2. The salesperson leaves the super-pixel at this portal
3. The salesperson enters and leaves the super-pixel at this portal
4. One agent enters the super-pixel at this portal
5. One agent leaves the super-pixel at this portal
6. None of the participants use this portal
Note that the conditions defining a portal-limited-solution assure that these six cases cover all
possible tour portions induced by all portal-limited-solutions (here we use the fact that two agents
do not happen to reach the same portal, as they start at pixel centers, their tours do not cross and
they end up at pixel centers). Also note that not all the 4m-size lists represent a valid portion of
some portal-limited-solution (they may represent non-matching numbers of entrances and exits
of the salesperson/agents, agents staying at a super-pixel which the salesperson does not visit,
or two agents leaving the same pixel). We use the term valid-list for a list that represents a
collection of tours that can be extended to some portal-limited-solution (and this validity can
easily be checked in O(m)time). Clearly, there are at most 64m = nO(1/ε) (valid-)lists. Finally,
note that the salesperson’s paths can intersect only at her entrance or exit points. Hence, given a
valid-list, pairings of the participants’ entrance and exit points can be found as in the algorithm
of Arora, and there are nO(1/ε) options for such pairings [Aro98].
We now describe the construction in a bottom-up manner. Consider a pixel. Each valid-list
of the pixel may fall into one the following three categories:
1. There is no agent in the pixel and the salesperson may visit the pixel one or more times.
2. There is one agent in the pixel. If the salesperson visits the pixel they meet at the pixel’s
center.
3. Two or more agents visit the pixel. The salesperson also visits the pixel. In one of the visits
she arrives at the center of the pixel. In this case, each agent travels along a straight line
32
Cooperative TSP
from a portal of the pixel to the center of the pixel. Alternatively, an agent’s route may be
an empty route if the agent is already located at the center of the pixel.
In each case, the computation of the minimal cost for each valid-list of the pixel can be done in
polynomial time. For each valid-list, the minimal cost is kept in the table, along with the pairing
for which this minimum is obtained.
We now turn to the computation of the table’s entries for the super-pixels of level i, assuming
all valid-lists of super-pixels of level i + 1 were computed. Let S be a level i super-pixel and
consider a list of 1, . . . , 6 values for its portals. The list already fixes the entrances and exits
on the boundary of S. The super-pixel S contains four level i + 1 super-pixels, which have four
boundaries internal to S, with a total of at most 4m more portals. Each of these portals may
be used in one out of the six ways, giving rise again to nO(1/ε) possibilities. The cost for each
possibility can be computed by using the values for the four i + 1 level super-pixels previously
obtained. Thus, we can find the minimal cost that corresponds to each list in nO(1/ε) time. For
each valid-list, we keep in the table which combination of uses of the internal boundaries provided
the minimal cost.
For the top-level super-pixel (the bounding-box) we may only consider the list for which
neither the salesperson nor an agent visit a portal. The last table update of level 0 produces the
cost of a minimal portal-limited-solution. We can reconstruct the solution itself, since for each
valid-list we kept the pairings and internal-boundary uses that had minimal cost.
The proof of Lemma 3.4 mainly follows arguments from the PTAS of Euclidean TSP [Aro98],
and is brought here for completeness.
Proof. (of Lemma 3.4)
Let π be an optimal solution. For every a, b ∈ {0, 2, . . . , L/2 − 2} denote by πab the (a, b)-shift
of π. We have to show, given a randomly chosen a and b, how to change πab to a portal-limitedsolution such that the expected increase in cost would be at most ε · OP T .
We refer to the axis-parallel lines of the form x = 2k or y = 2k, where k is an integer, as even
grid lines. Note that all portals are located on even grid lines.
Suppose that in π, a participant travels along a segment that crosses an even grid line `. Let
a and b be two even numbers chosen uniformly at random from 0, 2, . . . , L/2 − 2. Denote by àb
the (a, b)-shift of `. Note that the probability (over the choices of a and b) that àb contains a
boundary of a level i super-pixel is 2i /(L/4). Following the choice of a and b, we replace the
segment traveled by the participant by two segments, so that the crossing of àb is at the closest
portal on àb . The corresponding increase in cost is bounded by the interportal distance on àb
3.2 Euclidean cTSP
33
which is (L/2i )/m. Thus, we may bound the expected increase in cost due to this crossing by
log
XL
i=1
The last inequality holds as m ∈
h
L 2i
4 log L
ε
=
≤ √ .
2i m L/4
m
2 2
√
√
8 2 log L 16 2 log L
,
.
ε
ε
0 which is obtained by replacing each segment of the π by two
Now, consider a solution of πab
ab
axis-parallel segments
Clearly, the number of even grid lines crossings in πab is at most the number of even grid lines
√
0 , which is at most
2 · OP T .
crossings in πab
By combining the last two arguments we obtain that the total expected increase of cost is at
most ε/2 · OP T . Thus, we showed how to obtain a solution with an expected total cost of at most
(1 + ε/2) · OP T , which satisfies condition (1) of the portal-limited-solution definition.
Now we may remove self-intersections by “short-cutting”. In addition, if a portal is used more
than twice, we can keep “short-cutting” on the two sides of the portal until the portal is used
at most twice. (If this introduces additional self-intersections, they can also be removed.) The
obtained solution has an expected total cost of at most (1 + ε/2) · OP T and it satisfies conditions
(1) and (2) of the portal-limited-solution definition.
Note that changing the solution by moving each meeting point between an agent and the
salesperson to the nearest pixel center, increases the cost by at most O(n) = O(OP T /n). Additionally, note that if in our solution two (or more) agents happen to meet, then they may cease
to move. This holds, since in such a case the salesperson may come to meet the agents (and
return) without increasing the total cost. Combined with the previous argument, we obtain that
we can change the solution to also satisfy conditions (3) and (4) of the portal-limited-solution
definition, without increasing the total cost by more than O(1/n) · OP T . The latter cost is less
than ε/2 · OP T , for a sufficiently large n.
Thus, we obtained a portal-limited-solution which has an expected total cost of at most
(1 + ε) · OP T . Therefore, the proof is complete.
3.2.1.2
Min-Sum Sales Euclidean-cTSP
For this problem we obtain a 5/3 + ε approximation, and improve the ratio to 3/2 + ε for its
planar version (for an arbitrarily small ε > 0). We also prove NP-hardness, even for the planar
case.
34
Cooperative TSP
Lemma 3.5. Consider an instance of Min-Sum Sales Euclidean cTSP where there is no more
than one participant in any single point, and there are no three participants on the same straight
line. Solving it is equivalent to finding a bounded-degree minimum-spanning-tree, spanning the
initial locations of the participants, where the degree-bound is 1 for the salesperson’s tree-node and
3 for all the other nodes.
Proof. Consider an optimal solution for Min-Sum Sales Euclidean cTSP. Clearly, the participants move in straight lines between the initial locations of agents, to sell them the goods. We
assume w.l.o.g. that all the intersections between the participants’ routes occur at initial locations
of agents (if the routes of two agents intersect at another location, we can switch between them
and thus lower the cost of the solution). We can also assume w.l.o.g. that any initial location of
an agent is only visited once in an optimal solution.
Thus, in an optimal solution, the collection of the routes used by the participants forms a
spanning tree of their initial locations. The degrees of the spanning tree are bounded by 3, since
at most one participant enters a tree-node and at most two leave it. The node corresponding to
the salesperson must have degree 1.
On the other hand, any such bounded-degree spanning-tree produces a solution for our problem. Such a solution can be obtained by simply directing the edges of the spanning-tree from
parent to child and letting the participants follow these directed edges, starting with the salesperson (such that a single participant traverses each tree-edge). Therefore, finding a minimumspanning-tree which satisfies these degree-constraints is equivalent to solving our problem in this
case.
Corollary 3.6. Min-Sum Sales Euclidean cTSP can be approximated within 5/3 + ε, for any
ε > 0.
Proof. We slightly perturb the input locations, such that they satisfy the conditions of Lemma 3.5.
Khuller et al. [KRY96] showed that a minimum-spanning tree in any fixed-dimension Euclidean
space can be modified to satisfy the degree constraints we require (1 for a pre-specified node and
3 for the others), while increasing its weight by a factor of at most 5/3. Thus, the Corollary
follows.
For the planar case, we manage to improve the approximation ratio to 3/2 + ε:
Theorem 3.7. Min-Sum Sales Planar-cTSP can be approximated within 3/2 + ε, for any
ε > 0.
3.2 Euclidean cTSP
35
Algorithm Sales-Bounded-MST:
1. Compute a minimum-spanning tree, spanning all the agents’ initial locations
(not including the salesperson), such that its degrees are bounded by 5.
2. Let p be the location of the salesperson, and let q be the location of the agent
closest to him. Let r be the location of one of the agents connected to q in the
above tree. Transform the subtree rooted at r into a tree with degree at most
3, such that the degree of r is 1.
3. Transform the subtree rooted at q, without the subtree rooted at r, into a tree
with degree at most 3, such that the degree of q is 1.
4. Connect q to r and connect q to p. Output the resulting tree.
Figure 3.2: A 3/2-approximation algorithm for a 3-bounded-degree tree that spans the
participants’ locations and has salesperson’s node of degree 1.
Proof. By slightly perturbing the initial locations of the participants, we can assume that the
assumptions of Lemma 3.5 hold (this increases the optimal cost by a factor of 1 + ε, for an
arbitrarily small ε > 0). We look for the optimal solution of this slightly perturbed input, i.e.,
we look for a minimum spanning tree satisfying the degree-constraints stated in Lemma 3.5. We
find an approximate solution using the algorithm Sales-Bounded-MST of Figure 3.2.
The first stage of the algorithm can easily be performed in polynomial time [MS92]. Note
that connecting p and q right after this stage would have yielded a minimum-spanning-tree which
spans all the initial locations. Stages 2 and 3 are performed using the 32 -approximation algorithm
of Khuller et al. [KRY96], which requires that the degree of each node will be at most 5 and
the degree of the root will be at most 4. Thus, each of these stages increases the weight of the
transformed subtree by a factor of at most 3/2 [KRY96].
So all in all, we obtain a spanning tree which satisfies the degree bounds and has a weight of
at most 3/2 times a minimum-spanning-tree, which also means at most 3/2 times the cost of an
optimal solution.
As was explained in Lemma 3.5, the participants can now follow the edges of this tree (rooted
at p with edges directed from parent to child), and form a solution whose cost is the cost of the
tree. Thus, the Theorem follows.
Claim 3.8. Min-Sum Sales Planar-cTSP is NP-hard.
36
Cooperative TSP
Proof. Finding the minimum-spanning-tree whose degree is bounded by 3 is NP-hard [PV84]. We
note that the proof of [PV84] holds even if it is guaranteed that no three points of the input lie on
a straight line (since the input points can be slightly perturbed in their construction). Requiring
that a certain node will be a leaf only makes the problem harder (by solving the problem for all
the possible locations of a leaf one can find the solution for the problem without this requirement).
Since according to Lemma 3.5 this bounded-MST problem can be easily reduced to our problem (with the same input, where the salesperson is at the point which should be a leaf), our
problem is NP-hard.
3.2.1.3
Min-Sum Full-Cooperation Euclidean-cTSP
A (1 + ε)-approximate minimum Steiner tree, spanning all the participants’ initial locations, can
be computed by using the PTAS of Arora [Aro98]. Clearly, by letting the salesperson tour the
Steiner tree we obtain a (2 + 2ε)-approximate solution for our problem. Thus,
Corollary 3.9. Min-Sum Full-Cooperation Euclidean-cTSP can be approximated within
2 + ε, for any ε > 0.
3.2.2
Min-Max Euclidean-cTSP
3.2.2.1
Min-Max Purchase Euclidean-cTSP
We first show that both the path and the roundtrip versions of Min-Max Purchase Euclidean-cTSP have a PTAS. We do so by manipulating the input instance such that it fits the
PTAS for the graph version of the problem (see Subsection 3.3.2.1).
Claim 3.10. The roundtrip and path versions of Min-Max Purchase Euclidean-cTSP
admit a PTAS.
Proof. Consider an instance of the path version. We assume, w.l.o.g. that the instance lies inside
[0, 1]2 and has an optimal cost of at least 1/2. Let m = d εn0 e, where n is the number of participants
and ε0 is a parameter to be determined later. We divide the unit square [0, 1]2 into m2 pixels. I.e.,
0
0
j j+1
j j +1
a pixel is a square of the form [ m
, m ] × [m
, m ], where j, j 0 = 0, 1, . . . , m − 1. We consider a
slightly changed input, where each participant is located in the center of its pixel. This instance
can be approximated using Coarse-Path(G, W, v, ε00 ) - the PTAS for the corresponding graph
variant of the problem (see Subsection 3.3.2.1) as follows. Let G be a complete graph, with the
m2 pixels as vertices. Let W (e), the weight of each edge, be the Euclidean distance between the
3.2 Euclidean cTSP
37
corresponding pixels’ centers. Let v be the pixel which contains the salesperson and let ε00 be an
arbitrary small constant (to be determined shortly).
The solution for the altered instance is amended into a solution for the original instance by
connecting the original location of a participant and the center of its pixel (each such connection
√
is of length at most
2
2m
√
=
2ε0
2n ).
Denote by OP T the optimal solution for the original instance, by OP T 0 the optimal solution
for G, by ALG0 the output of the PTAS for G, and by ALG the output of the whole algorithm.
Both OP T and OP T 0 consist of at most 2n segments (one for each agent and at most n for the
√ 0
√
2ε
salesperson). Therefore, OP T 0 is at most 2n · 2n
= 2ε0 larger than OP T . Similarly, ALG is
√
at most 2 2ε0 larger than ALG0 . Thus,
√
√
√
√
ALG ≤ ALG0 + 2 2ε0 ≤ (1 + ε00 )OP T 0 + 2 2ε0 ≤ (1 + ε00 )(OP T + 2ε0 ) + 2 2ε0
√
≤ (1 + ε00 )OP T + 4 2ε0
Therunning time of the algorithm is dominated by the running time of Coarse-Path. Thus,
(2/ε00 )+6
0
it is O (n/ε )
. Choosing ε0 = 1/ lg n and ε00 = ε − 12/ lg n, as 1/2 ≤ OP T we obtain
that ALG ≤ (1 + ε)OP T and the running time is O (n lg n)2/ε−(12/ lg n)+6 .
Exactly the same arguments yield a PTAS for the roundtrip version, with the same running
time.
3.2.2.2
Min-Max Sales Euclidean-cTSP
Claim 3.11. Min-Max Sales Euclidean-cTSP can be approximated within a factor of 3 both
for the path and the roundtrip versions of the problem.
Proof. We consider the complete graph whose vertices are the initial locations of the participants
and whose edge-weights are the distances. We solve the problem for that graph using algorithm
Hop-visit, described for graphs in Subsection 3.3.2.2. The same analysis holds here as well.
3.2.2.3
Min-Max Full-Cooperation Euclidean-cTSP
Similarly to Claim 3.11, we can obtain the approximation by considering the complete graph whose
vertices are the initial locations of the participants and whose edge-weights are the distances. We
use here the algorithm described in Subsection 3.3.2.3, and the same analysis holds here as well.
We thus have the following Corollaries:
38
Cooperative TSP
Corollary 3.12. Min-Max Full-Cooperation Euclidean-cTSP can be approximated within a factor of 4.
Corollary 3.13. The roundtrip version of Min-Max Full-Cooperation Euclidean-cTSP
can be approximated within a factor of 2.
3.2.3
Min-Makespan Euclidean-cTSP
In this Subsection we mainly present a simple PTAS for the roundtrip version of Min-Makespan
Sales Euclidean-cTSP. A PTAS for the corresponding Full-Cooperation problem, in both
the path and roundtrip versions, can be obtained by similar means.
In addition, we note that both the path and the roundtrip versions of the corresponding
Purchase problem, namely Min-Makespan Purchase Euclidean-cTSP, are polynomialtime solvable. This can be observed as follows. For both the path and the roundtrip versions
any optimal solution can be modified to an optimal solution in which all participants meet at a
single point. For the path version, the modification can be done by letting all the participants
meet the salesperson at the last point she visits. For the roundtrip version, denote the value
of an optimal solution by OP T . Then, the modification of the optimal solution can be done
by letting all the participants meet at the point where the salesperson resides at time OP T /2
(afterwards, all participants return to their initial location). Hence, for both the path and the
roundtrip cases, the single meeting point is the center of the enclosing sphere, and can thus
be found in polynomial time (see for example Megiddo [Meg83], Welzl [Wel91] and de Berge et
al. [dBvKOS00]).
A PTAS for the path version of Min-Makespan Sales Euclidean-cTSP has been obtained
by Arkin et al. [ABF+ 02]). We next present a PTAS for the roundtrip version of this problem.
3.2.3.1
The Roundtrip version of Min-Makespan Sales Euclidean-cTSP
This problem seems quite close in nature to the corresponding path version, and thus calls for
a similar PTAS. However, note that converting an optimal solution for the path version into a
solution for the roundtrip version (by simply letting all the participants return) only guarantees
a (2 + ε)-approximation for this problem. This is true since a participant’s way back may double
the makespan, while the roundtrip version may have a solution with the same makespan as
the path version. Thus, constructing a PTAS for the roundtrip version requires considering in
advance that the participants should return to their initial locations. We therefore use a different
approach than the one used by [ABF+ 02] for the path version of the problem.
3.2 Euclidean cTSP
39
We show the PTAS for the two dimensional case (see Figure 3.3). The generalization to any
fixed dimension is straightforward.
Theorem 3.14. The roundtrip version of Min-Makespan Sales Euclidean-cTSP admits
a PTAS. The running time of the PTAS is O(n + f (ε)), where ε > 0 is an arbitrarily small
constant, f (ε) depends only on ε, and n is the number of participants.
Proving this theorem requires some preparations and preliminary observations.
A constant approximation algorithm for the path version of this problem appears in [ABF+ 02].
The solution found by their algorithm is also O(1) times the diameter (the maximal distance
between any two points) of the input. One can adapt this approximation to the roundtrip
version simply by returning each participant to its origin. The cost of the resulting solution is
at most twice the original solution. Since an optimal solution to the path version costs less
than an optimal solution for the corresponding roundtrip version, this heuristic is a constant
approximation for the roundtrip version.
We assume, w.l.o.g. that the instance lies inside [0, 1]2 and has an optimal cost of at least
1/2. Let m = d1/εe. We divide the unit square [0, 1]2 into m2 pixels. I.e., a pixel is a square of
0
0
j j +1
j j+1
, m ] × [m
, m ], where j, j 0 = 0, 1, . . . , m − 1.
the form [ m
The PTAS for the Sales version relies on the next lemma:
Lemma 3.15. Let I be an instance of n participants with an optimal makespan of OP T . Then,
there exists an instance S ⊆ I with at most 3m4 participants, in which each non-empty pixel in I
is also non-empty in S and the optimal makespan of S is at most (1 + O(ε))OP T .
Proof. We may assume, w.l.o.g., that no two participants in I are located at the same point and
that no three participants lie on a straight line. Otherwise, we can perturb each participant’s
location by at most ε/n. An optimal solution to the perturbed instance has a cost of at most
OP T + O(ε) (as the cost increase per participant is at most
2ε
n ).
Since OP T ≥ 1/2 this cost is
less than (1 + O(ε))OP T .
We show how we can remove participants from I while keeping the cost of an optimal solution
to be at most OP T + O(ε). Let π be an optimal solution to I. We define the sales-tree of π to be
a directed graph in which the nodes are the locations of the participants and there is a directed
edge from u to v if a participant traveled from u to v in π. Since no two participants are located
at the same point and no three participants lie on a straight line the in-degree of every node is one
and the out-degree is at most two. We prune the sales-tree of π by iteratively removing leaves:
we remove a leaf u if there exists another node in the sales-tree which resides in the same pixel
as u. At the end of the process we are left with at most m2 /2 leaves, and at most m2 /2 nodes
40
Cooperative TSP
Makespan-Sales PTAS
1. For each pixel which contains more than 3m4 agents, arbitrarily select a subset
of 3m4 agents. Let P be the subset of the participants which contains all these
selected agents, as well as all the agents of the other pixels (that contained at
most 3m4 agents), and the salesperson.
2. Enumerate over subsets S ⊆ P , of size up to 3m4 , which include the salesperson
and contain a representative from each non-empty pixel. For each such subset
S:
(a) Find an optimal solution for S by conducting an exhaustive search.
(b) In each non-empty pixel apply a constant-approximation to all original
participants of the pixel, where the salesperson is a representative of the
pixel.
(c) Extend the partial solution of S to a solution for the original instance:
when all the participants in S return to their pixels - simultaneously perform the solution found in step 1(b).
3. Return the minimal cost solution found
Figure 3.3: A PTAS for the roundtrip version of Min-Makespan Sales EuclideancTSP. The parameter m is assumed to be d1/εe.
of degree 3 (in-degree plus out-degree). Note that the makespan of an optimal solution for the
new instance, denoted π0 , is at most OP T . We now further decrease the number of participants
by pruning some of the degree-2 vertices. We call a maximal set of participants along a path in
which all the nodes are of degree 2 a chain. Clearly, each chain ends with a degree 3 node or a
leaf. Hence, there are at most m2 chains. For each chain, and a pixel it intersects with, we intend
to keep at most two nodes (participants). All the other nodes are removed from the chain. For a
given pixel and a chain, the two participants that we keep are the first and the last (of this chain,
inside the pixel) who receive the goods. We call such nodes a beginner node and an ender node,
respectively. Note that, we are left with at most 2 · m2 participants per chain, giving rise to at
most 2m4 nodes of degree 2.
The new instance constructed, denoted S, has at most 3m4 participants. We next show that
Claim 3.16. There exists a solution πS for S of cost at most OP T (1 + O(ε)).
Proof. Recall that π0 (an optimal solution after pruning the leaves) is of cost at most OP T . We
3.2 Euclidean cTSP
41
construct the solution πS from π0 as follows: Each participant of a beginner node travels along
the corresponding original chain until it reaches the corresponding ender node, and then travels
back to its starting location. All other participants travel along the same route they travel in
π0 . Thus, they arrive to their original location by the time OP T . Beginner participants may be
delayed by the time it takes to travel from the corresponding ender node back to their original
√
location. This is at most the time it takes to cross a pixel which is at most 2ε. Thus, the cost
of an optimal solution to S is at most (1 + O(ε))OP T .
This concludes the proof of Lemma 3.15.
The correctness of the PTAS algorithm for the roundtrip version of Min-Makespan Sales
Euclidean-cTSP can now be deduced:
Proof. (of Theorem 3.14) Let π be an optimal solution for the instance I and let S ∗ ⊆ I be an
instance that satisfies the condition of Lemma 3.15. The cost of an optimal solution to S ∗ is
(1 + ε)OP T . The subset of participants S ∗ is not necessarily included in the enumeration of our
algorithm. However, our enumeration does include a subsets S, such that |S| = |S ∗ | and S and
S ∗ have exactly the same number of agents in each of the pixels. Clearly, the agents in S can
simulatneously move to the positions of the agents in S ∗ , in O() time. Thus, the cost of an
optimal solution for S, which is computed at stage 1(b) of our algorithm, is (1 + O(ε))OP T . The
additional cost produced at stage 1(c) is at most a constant times the diameter of the pixel, which
is O(ε). Note that this is an additive O(ε) increase of the makespan, as after all the participants
in S return to their pixels the delivery to the other participants is done in parallel. Hence, the
total cost of the solution produced by our algorithm is at most (1 + O(ε)) times the cost of π.
4)
Finally, note that there are less than O(m4 )O(m
4)
= O(1/ε)O(1/ε
sets of participants to
enumerate on. For each such subset S, a solution is a sequence of at most 2|S| − 1 moves. This
follows as in each move either a participant receives the delivery or a participant returns to its
original location. In any case, each move can be represented as a pair of two of the original input
locations. Hence, for a given subset S, the number of solutions the algorithm enumerates on is at
most
4 O(m4 ) ! O(1/ε4 )
|S| 2|S|−1
m
1
.
=O
=
ε
2
2
Thus, the algorithm is a PTAS and runs in time O n +
1
1 O( ε4 )
ε
.
42
Cooperative TSP
3.3
cTSP in Graphs
In this section we present the algorithmic and hardness results for cTSP in graphs.
3.3.1
Min-Sum cTSP
We start by considering cTSP with the Min-Sum objective. For the path versions, we provide
simple approximation algorithms and hardness results for each cooperation mode.
For the roundtrip versions, the corresponding Purchase, Sales and Full-Cooperation
problems are all equivalent to the classical metric-TSP, as explained in Claim 3.1, and thus
have the same approximation and hardness results. We therefore have the following corollary:
Corollary 3.17. The roundtrip versions of Min-Sum Purchase cTSP, Min-Sum Sales
cTSP and Min-Sum Full-Cooperation cTSP can all be approximated within 3/2, and cannot
be approximated within 131/130, unless P 6= N P .
These observations do not hold for the other objectives, in which there is also a significant
difference between the different cooperation-modes.
We begin with the approximability results for the various path version:
Claim 3.18. Under the Min-Sum objective, Purchase and Full-Cooperation cTSP can
be approximated within 2 + ln 3. If each vertex contains a participant, the approximation ratio
improves to 2. Min-Sum Sales cTSP can be approximated within 2.
Proof. For the first two problems, we simply find an approximate minimum Steiner-tree that
spans the vertices which contain participants, and the salesperson visits all the agents by touring
this tree (e.g., in an “infix-order”). The total distance traveled is twice the tree’s weight. We
use the approximation algorithm of [RZ00] for the minimum Steiner-Tree problem, which has an
approximation ratio of 1 + (ln 3)/2 (' 1.55). Therefore, the distance traveled is at most (2 + ln 3)
times the weight of the minimum Steiner-tree.
On the other hand, the edges used by any solution to these problems must form a connected
subgraph which spans the vertices that contain participants (since all the agents receive the
goods). This means that the total distance traveled is at least the minimum Steiner-Tree weight.
Therefore, the simple algorithm described has an approximation ratio of 2 + ln 3. If each vertex contains a participant, then a minimum-spanning-tree can be computed exactly. Thus, the
approximation ratio in this case is 2.
For Sales cTSP it is again sufficient to compute a minimum-spanning-tree, since the goods
can be delivered to an agent only in the original vertex of that agent.
3.3 cTSP in Graphs
43
We next provide hardness results for each cooperation-mode. As in the Euclidean case, the
Purchase version is NP-hard, since the classical path-TSP [Hoo91] can be reduced to an instance of Min-Sum Purchase cTSP. The reduction is again done simply by replacing each
customer with three agents. Thus, the salesperson is the only participant who moves. Like TSP,
the path-TSP problem has a 3/2 approximation when the triangle inequality holds [Hoo91].
Therefore, improving the approximation ratio for our problem strictly below 3/2 will also improve the approximation for path-TSP.
We next address the NP-hardness of Min-Sum Sales cTSP.
Claim 3.19. Min-Sum Sales cTSP is NP-hard.
Proof. We use a reduction from path-TSP [Hoo91]. Recall that an instance of path-TSP consists
of a complete weighted undirected graph, G = (V, E), in which the weight function satisfies the
triangle inequality, and a vertex v ∈ V , in which the salesperson is located. A solution is a
Hamiltonian path which has v as one of its endpoints. The goal is to find a solution of minimal
weight.
Given an instance of path-TSP, we construct an instance of Min-Sum Sales cTSP as
follows. For each vertex u 6= v, we add a vertex u0 , and connect it to u by an M -weighted
edge, where M is twice the sum of the edge weights of G plus 1. We denote this new graph by
G0 = (V 0 , E 0 ). Each vertex of G0 contains a participant, and the participant in v is defined to be
the salesperson.
Clearly, if the optimal path-TSP solution is of length C, then there is a solution for the new
problem with total length (n − 1)M + C.
On the other hand, assume there is an optimal solution for the new problem with a total cost
of (n − 1)M + C. Clearly, the agents in new vertices don’t move in such a solution (since it already
costs at least (n − 1)M to reach them, and if such an agent moves the cost is increased by at least
M ). We prove that there is an optimal solution for that problem in which agents adjacent to new
vertices only move to the new vertex adjacent to them, and therefore the salesperson visits all the
vertices of G by traversing a path of length C (this path is simple since the triangle inequality
holds in the original graph G).
Assume this is not true. Hence, there is an agent who travels to a vertex which is not the
new vertex adjacent to it. Let the agent who started at vertex u be the first such agent which
the salesperson meets.
Clearly, some agent must visit u0 . We can assume w.l.o.g. that this agent receives the goods
through the agent of vertex u (not necessarily directly from him), since otherwise we can simply
“switch names” between the agent of vertex u and the salesperson when they meet. It is easy
44
Cooperative TSP
to see that by switching-names between agents when they meet, we can obtain a solution with
the same cost, in which the agent of vertex u is the agent who returns to u and moves to u0 .
Therefore, the salesperson could have done the tour of that agent by himself and return to u, and
the agent of vertex u could go immediately to u0 , without affecting the cost of the solution or the
visited agents.
This argument can also be applied to each of the next agents which the salesperson meets in
the given optimal solution. Therefore, there is an optimal solution in which these agents only
move to new vertices, as required. Thus, there is a Hamiltonian-Path in G which starts at v and
has total length C, and the proof is completed.
For Min-Sum Full-Cooperation we have:
Theorem 3.20. Min-Sum Full-Cooperation cTSP is APX-hard.
Proof. We use a reduction from a variant of Set-Cover, in which each element appears in exactly
k sets and each set is of size d. We call this variant (k, d)-Set-Cover. We rely on the following
theorem of [DGKR05]:
Theorem 3.21. [DGKR05] For every k > 2, ε > 0 and a sufficiently large n, there exists a
positive integer d, such that the following holds: Given an instance of (k, d)-Set-Cover with n
elements and m sets, it is NP-hard to decide whether there exists a solution of size
m
k−1−ε
or every
solution is of size at least (1 − ε)m.
Let C be an instance of (k, d)-Set-Cover, with k and d values for which the last Theorem
holds. C is a collection of m subsets of size d of a finite set S (|S| = n), such that each element
of S appears in exactly k subsets. We construct the following instance of our problem. Let the
graph G = (V, E) be constructed as follows. We have a vertex vc for every set c ∈ C, a vertex vs
for every element s ∈ S, and two other vertices u, v. Namely,
V = {u, v} ∪ VC ∪ VS ,
where
VC = {vc | c ∈ C},
VS = {vs | s ∈ S}.
The edge-set E is defined as follows: every vertex vc ∈ VC is connected to v, there is an edge
between vs ∈ VS and vc ∈ VC iff s ∈ C, and u is connected to v. Namely,
E = {(u, v)} ∪ {(v, vc ) | c ∈ C} ∪ {(vc , vs ) | s ∈ c, c ∈ C}.
3.3 cTSP in Graphs
45
Figure 3.4: An illustration of the graph construction described in the proof of Theorem 3.20.
The weight of each edge is 1, except for the edge (u, v), whose weight is 0. Each vertex vs ∈ VS
m
contains one agent, v contains d k−1−ε
e − 1 agents, and u contains the salesperson. Let B be the
instance constructed for our problem.
Claim 3.22 (Completeness). If there is a solution for C of size at most
solution for B of cost at most
m
d k−1−ε
e
m
k−1−ε ,
then there is a
+ n.
m
Proof. Let C 0 be the solution for C of size at most d k−1−ε
e. The solution for B is as follows.
m
The salesperson moves to v, and then d k−1−ε
e participants traverse from v to VC 0 where VC 0 =
{vc | c ∈ C 0 }. Each of the n agents at VS moves to a neighbor in VC 0 (as C 0 is a cover, every vertex
m
in VS has such a neighbor). Thus, the total cost of the solution for B is at most d k−1−ε
e + n.
Claim 3.23 (Soundness). If every solution for C is of size at least (1 − ε)m then every solution
for B is of cost at least n + (1 − ε)m.
Proof. Note that there is an optimal solution in which every agent in VS makes at least one step.
This holds, since otherwise another participant has to traverse an edge adjacent to that agent, so
the solution can only be cheaper if that agent from VS moves towards the other participant.
As every solution for C is of size at least (1 − ε)m, at least (1 − ε)m of the vertices of VC
are populated after the agents in Vs make one step. Therefore, at least (1 − ε)m more steps are
needed for that optimal solution. Thus, every solution to B is of cost at least n + (1 − ε)m.
46
Cooperative TSP
Coarse-Path(G = (V, E), W, v, ε):
1. For each ordered subset V 0 ⊆ V of size 1 + b1/εc or less, which starts with v.
(a) For each u ∈
/ V 0 that contains an agent, find its distance to a closest vertex
0
in V . Denote the maximal distance found by MaxDist(V’).
(b) Compute the sum of distances between pairs of consecutive vertices in V 0 ,
and denote it by Length(V’).
(c) Let Cost(V’) be the maximum of Length(V’) and MaxDist(V’).
2. Pick the ordered subset V 0 for which Cost(V’) is minimal.
3. Return the following solution: The salesperson follows the shortest paths between the consecutive vertices of V 0 . Each of the agents meets the salesperson
at a closest vertex to that agent in V 0 . The salesperson waits for all the agents
who come to a certain vertex before moving to the next vertex.
Figure 3.5: A PTAS for Min-Max Purchase cTSP.
By the completeness and soundness claims we obtain that it is NP-hard to approximate the
problem to within
n+(1−ε)m
m
n+ k−1−ε
+1 ,
which is about
d+(1−ε)k
d+1
(as m =
kn
d ).
As k > 2, the problem is
APX-hard.
3.3.2
Min-Max cTSP
We first present a simple PTAS for the Purchase version of this problem, and then prove that
it has no FPTAS, assuming P 6= NP. For the other cooperation-modes, we present constant lower
bounds on the approximation-ratio, assuming P 6= N P . We also provide algorithms that find
constant factor approximations for these problems, which are tight in one case, and are at most
twice the lower bounds in the other cases. The results for the roundtrip versions resemble the
results for the corresponding path versions.
3.3.2.1
Min-Max Purchase cTSP
We start by presenting the PTAS for the Purchase version, which appears in Figure 3.5
(algorithm Coarse-Path).
Theorem 3.24. Algorithm Coarse-Path is a PTAS for Min-Max Purchase cTSP.
3.3 cTSP in Graphs
47
Proof. Clearly, the Min-Max cost of the solution returned by the algorithm is the minimal
Cost(V 0 ) of the subsets it considers. We show that one of these subsets has Cost(V 0 ) of at most
(1 + ε) times the optimum.
Consider an optimal solution to the problem π, in which the cost is OP T . Choose a subset of
the vertices of the path traveled by the salesperson in the following way. Start with vertex v, and
then choose a vertex iff its distance from the previous vertex chosen is at least ε · OP T . Clearly,
at most 1/ε vertices are selected. Denote this subset by V 0 . Note that Length(V 0 ) ≤ OP T .
For each vertex u ∈
/ V 0 that contains an agent, there is a vertex in V 0 at a distance of
at most (1 + ε) · OP T . This holds, since for each vertex w visited by the salesperson in π, V 0
contains a vertex at a distance of at most ε·OP T from w. Thus, M axDist(V 0 ) ≤ (1+ε)OP T , and
Cost(V 0 ) ≤ (1 + ε)OP T . Therefore, Algorithm Coarse-Path indeed finds a (1 + ε)-approximate
solution. The running-time of the algorithm is O(nb1/εc+3 ), since it enumerates over ordered
subsets of vertices of size at most b1/εc, and the required computation for each ordered subset
takes at most O(n3 ) time (if we assume that the graph is complete then this is O(n/ε)). Thus,
Coarse-Path is a PTAS.
We similarly have a PTAS for the roundtrip version of the problem. Simply let all the
participants return to their initial vertex at the end of Algorithm Coarse-Path, and compute
the costs accordingly. It is easy to see that the arguments used for the path version hold here as
well. Thus, we have:
Corollary 3.25. The roundtrip version of Min-Max Purchase cTSP admits a PTAS.
We next present the tight hardness results for these versions.
Claim 3.26. Min-Max Purchase cTSP has no FPTAS, unless P = N P .
Proof. We show a reduction from the Hamiltonian Path problem, where a given vertex v ∈ V
must be an endpoint of the path. Given an input to that problem, G = (V, E), v ∈ V , we
construct an instance of our problem in the following way. For each u ∈ V , we add a vertex u0
and an edge (u, u0 ), with a weight of n − 1 (the weights of the original edges remain 1). We locate
the salesperson at v, and we locate an agent at each of the newly added vertices. It is easy to see
that the instance of the Hamiltonian Path decision problem is a “yes” instance iff the value of
the optimal solution of the new instance is n − 1. Thus, our problem is strongly N P − hard (the
n − 1 weight used in the reduction is obviously polynomial in the input-size), and therefore has
no FPTAS.
Claim 3.27. The roundtrip version of Min-Max Purchase cTSP has no FPTAS, unless
P = NP.
48
Cooperative TSP
Hop-visit(G = (V, E), W, v):
1. Let G0 = (V 0 , E 0 ) be a weighted complete graph, where V 0 ⊆ V is the set of
vertices which contain participants, and the edge-weights are the corresponding
distances in G.
2. Compute a minimum-spanning-tree T of G0 , and root it at the salesperson’s
vertex v.
3. The salesperson visits an arbitrary child, and doesn’t move any further.
4. When an agent receives a delivery:
(a) If the agent has a sibling in T who has not received the delivery, then the
agent visits such a sibling and one of that sibling’s children.
(b) Otherwise, the agent visits a child of the sibling which was visited first (a
child of the “eldest” sibling of that agent), if such a child exists.
Figure 3.6: A 3-approximation algorithm for Min-Max Sales cTSP
Proof. Similarly to the proof of Claim 3.26, we apply a reduction from Hamiltonian Cycle.
Given an input G = (V, E), we locate the salesperson in one of the vertices, v. Additionally, for
each u ∈ V , we add a vertex u0 , connected by an edge (u, u0 ) with a weight of n/2 (the weights of
the original edges remain 1). All the newly added vertices contain an agent. It is easy to see that
the instance of the Hamiltonian Cycle problem is a “yes” instance iff the value of the optimal
solution of the new instance is n. Thus, our problem is strongly NP-hard and has no FPTAS.
3.3.2.2
Min-Max Sales cTSP
In this Subsection we present a 3-approximation algorithm for both the path and roundtrip
versions of the problem. We later prove that these problems cannot be approximated within a
factor of less than 2 and 3/2, respectively (unless P = N P ).
The simple constant approximation algorithm for Min-Max Sales cTSP is presented in
Figure 3.6.
Theorem 3.28. Min-Max Sales cTSP is 3-approximable.
Proof. We prove that Algorithm Hop-visit is a 3-approximation algorithm for this problem.
Clearly, all the agents are visited. Each participant traverses at most three edges of the MST,
3.3 cTSP in Graphs
49
which means that the cost of the solution is at most thrice the weight of the heaviest edge of the
MST.
On the other hand, consider an optimal solution, and define G00 = (V 0 , E 00 ), such that (u1 , u2 ) ∈
E 00 iff the participant from u1 sold the goods to the participant from u2 , or vice versa. Let the
weight of (u1 , u2 ) ∈ E 00 in G00 be the distance between u1 and u2 in G. The optimal cost is clearly
at least the weight of the heaviest edge in E 00 , since selling to an agent requires traveling to this
agent’s vertex.
Note that G00 is a connected subgraph of G0 . It is well-known that an MST is lexicographically
minimal, i.e., its heaviest edge is not heavier than that of any other spanning-tree or spanningconnected-subgraph. Therefore, the cost of the solution found by the above algorithm is at most
thrice the cost of an optimal solution.
A similar argument holds for the roundtrip version. We use algorithm Hop-visit, and then
let each participant return to its original vertex (using the shortest path). Clearly, the Min-Max
value is at most 6 times the weight of the heaviest edge of the MST of G0 . On the other hand,
the optimal cost is at least twice the weight of the heaviest edge of G00 (since selling to an agent
requires reaching him and then returning back). Thus, we have:
Corollary 3.29. The roundtrip version of Min-Max Sales cTSP is 3-approximable.
We next show lower bounds on the approximability of both the path and the roundtrip
version of Min-Max Sales cTSP.
Claim 3.30. Min-Max Sales cTSP cannot be approximated better than 2, unless P 6= N P .
Proof. The reduction is from the Hamiltonian-Path problem where one endpoint of the path,
vertex u, is specified in the input.
Given an instance of that Hamiltonian-Path problem, an unweighted undirected graph
G = (V, E) and a vertex u, we construct an instance for our problem by simply locating the
salesperson at u and putting an agent at each of the other vertices.
If there is a Hamiltonian path in G which starts at u, then there is a solution for the new
problem where the Min-Max value is 1, as follows. The salesperson moves to the next vertex in
the path, sells to the agent there, and doesn’t move any further. Then, each agent moves to the
next vertex on the path, sells the goods to the agent there, and also doesn’t move any further.
Thus, all the agents are visited, and the maximal distance traveled is 1.
On the other hand, if there is a solution with Min-Max value 1, then the salesperson and the
agents each make at most one step and stop (i.e., they traverse at most one edge each). Since
50
Cooperative TSP
all the agents are visited in a solution, each of the first |V | − 1 steps must have visited a vertex
which hadn’t been visited before. Thus, following the steps in this sequence gives a Hamiltonian
path which starts at u.
It is therefore NP-hard to distinguish between an instance which has a solution with MinMax value 1 and an instance which only has solutions with Min-Max values 2 or more. Thus,
it is NP-hard to approximate the value of the optimal solution within a factor lower than 2.
The reduction for the roundtrip version of the problem is identical. However, here a “yes”
instance implies a cost of 2, and it is NP-hard to distinguish between an instance which has a
solution of cost 2 and an instance which only has solutions of cost 3 or more. We thus have:
Claim 3.31. The roundtrip version of Min-Max Sales cTSP cannot be approximated better
than 3/2, unless P 6= N P .
3.3.2.3
Min-Max Full-Cooperation cTSP
The Min-Max Full-Cooperation cTSP problem allows only constant-factor approximations.
We prove a lower bound of 2 on the approximation ratio for both the path and roundtrip versions of the problem. Additionally, we provide a simple algorithm which obtains a 4-approximate
solution for the path version and a tight 2-approximate solution for the roundtrip. We start
by considering the special case in which each vertex contains at least one participant.
Claim 3.32. Min-Max Full-Cooperation cTSP is 2-approximable if each vertex contains at
least one participant.
Proof. Compute an MST rooted at the salesperson’s vertex, and let one agent from each vertex
move to her parent’s vertex, and return after receiving the goods (the agents initially located
at the leaves don’t need to return). The maximum distance traveled by any of the agents is at
most twice the weight of the heaviest edge in the MST. On the other hand, any solution to the
problem forms a spanning-connected-subgraph, and its Min-Max value is at least the weight
of the heaviest edge in that subgraph. As we noted before, since the MST is lexicographically
minimal, its heaviest edge is not heavier than that of any other spanning-connected-subgraph.
Hence, the algorithm is a 2-approximation algorithm.
Claim 3.33. Min-Max Full-Cooperation cTSP is 4-approximable.
3.3 cTSP in Graphs
51
Proof. The proof is similar to the proof of the approximation for the Min-Max Sales version.
We define the weighted complete graph G0 = (V 0 , E 0 ), where V 0 is the set of vertices which contain
participants, and the edge-weights are the distances between these vertices in the original graph.
We now perform the same algorithm as in the last proof: We compute an MST rooted at the
salesperson’s vertex, one agent from each vertex moves to the vertex of her parent, and she returns
after receiving the goods. The maximum distance traveled by any of the agents is again at most
twice the weight of the heaviest edge in the MST.
On the other hand, consider an optimal solution, and define G00 = (V 0 , E 00 ), s.t. (u, v) ∈ E 00
iff participants from u and v meet during that solution (the weight is again the distance between
them). The optimal Min-Max value is clearly at least half the weight of the heaviest edge in E 00 ,
since a meeting of two participants requires that at least one of them traversed half the distance
between them. Also, G00 is clearly a spanning-connected-subgraph of G0 , and its heaviest edge has
at least the cost of the heaviest edge of the MST of G0 found by the above algorithm. Therefore,
this algorithm achieves an approximation ratio of 4.
Note that the simple algorithm described in the last proof also solves the roundtrip version
of the problem (with the same cost). On the other hand, the bound on the cost of the optimum is
doubled for the roundtrip version (if two participants meet and return then one of them must
travel at least the distance between them). Thus:
Corollary 3.34. The roundtrip version of Min-Max Full-Cooperation cTSP is 2-approximable.
We now turn to presenting hardness results.
Theorem 3.35. Min-Max Full-Cooperation cTSP cannot be approximated better than 2,
unless P = N P .
Proof. We prove this by a reduction from the Set-Cover problem.
The Reduction: Let (C, k) be an instance of Set-Cover, where C is a collection of subsets of
a finite set S, and k is an integer. It is NP-hard to decide whether there is a set cover for S of
size at most k, i.e., a subset C 0 ⊆ C such that every element in S belongs to at least one member
of C 0 .
We use the same construction as in the hardness proof for the Min-Sum objective (Theorem 3.20), except that v contains only k − 1 agents.
Claim 3.36 (Completeness). If (C, k) is a “yes” instance of Set-Cover, then there is a solution
for our problem with maximum-distance 1.
52
Cooperative TSP
Proof. Let C 0 ⊆ C be a set cover of S of size |C 0 | ≤ k. Let VC 0 = {vc | c ∈ C 0 }. Then the
salesperson moves to v, and the k participants now populating v go to the vertices of VC 0 (at least
one participant to each vertex); the agents populating the vertices VS move to VC 0 as well (each
to a closest vertex in VC 0 ). As C 0 is a set cover, each of the agents at VS has a vertex vc ∈ VC 0 at
distance 1. Hence this scheme ends in one step.
Claim 3.37 (Soundness). If there is a solution with maximum-distance lower than 2, then (C, k)
is a “yes” instance of set-cover.
Proof. As all edges are of length 1, every agent ends up at some vertex in VC . Let VC 0 be the set
of those vertices. Clearly |VC 0 | ≤ k as there are originally only k − 1 agents at v and a salesperson
in u, and every other agent has to meet one of them. Thus, C 0 = {c | vc ∈ VC 0 } is a set cover for
S of size at most k, hence (C, k) is a “yes” instance of Set-Cover.
Corollary 3.38 (Hardness of Approximation). It is NP-hard to distinguish between an instance
of Min-Max Full-Cooperation cTSP with value 1 and an instance with value 2. Hence, this
problem is NP-hard to approximate to within any factor smaller than 2.
The reduction for the roundtrip version is identical. By using the same considerations, a
set-cover of size k or less exists iff there is a solution with Min-Max value 2 to the new problem.
Note that there can be no solution with Min-Max value 3, since there are no triangles in the
constructed graph. We thus have:
Corollary 3.39. The roundtrip version of Min-Max Full-Cooperation cTSP cannot be
approximated better than 2, unless P = N P .
3.3.3
Min-Makespan cTSP
The Min-Makespan objective is the most diverse out of the three. The Purchase problem
has a polynomial-time solution for both the path and the roundtrip versions. The FullCooperation version can be approximated within a ratio of 2, and this cannot be improved,
√
unless P 6= N P . For the Sales version, only an O( log n) approximation is known [KLS04],
while the lower bounds for that version are smaller than 2 (5/3 for the path version, shown by
[ABF+ 02], and 5/4 for the roundtrip version, which we show below).
3.3 cTSP in Graphs
3.3.3.1
53
Min-Makespan Purchase cTSP
Claim 3.40. Min-Makespan Purchase cTSP can be solved in O(mn + n2 log n) time.
Proof. We observe that there is an optimal solution in which all the agents meet the salesperson
at a single vertex. This holds, since the value of a solution does not change if each agent that
meets the salesperson joins her in her journey. Thus, they could have all met at the last vertex
visited by the salesperson in that solution, without increasing the completion-time. Specifically,
this argument is true for the optimal solution. Hence, an optimal solution can be found simply by
computing all-pairs-shortest-paths in the graph and finding the vertex whose maximal-distance
from any of the participants is minimal. This takes the above stated time using Dijkstra’s algorithm (e.g. [CLRS01]).
Claim 3.41. The roundtrip version of Min-Makespan Purchase cTSP can be solved in
O(mn + n2 log n) time.
Proof. The idea of this proof is similar to the idea of the previous one, but it is slightly more
involved. We first show that there exists an optimal solution in which all the agents meet the
salesperson either in a single vertex or in two adjacent vertices.
Consider an optimal solution of cost OP T . Let u be the last vertex visited by the salesperson
until time OP T /2 in that solution, and let v be the first vertex she visited after that time. Clearly,
all the agents that the salesperson met before leaving u can join her in her way to u and then
return, without increasing the makespan.
We observe that all the agents which the salesperson met after leaving u can come to meet her
at v and return to their initial vertex before time OP T . Let the meeting of such an agent with the
salesperson occur at vertex w. Then clearly that agent’s travel to w plus the salesperson’s travel
from v to w take less then OP T /2 time. Thus, all these agents can reach v before the salesperson
(in less than OP T /2 time). They can return to their initial vertices before time OP T , since they
can join the salesperson’s tour until the vertex where they originally met, and then return to their
initial vertex just like in the original optimal solution.
This means that an optimal solution can be found by computing all-pairs-shortest-paths and
enumerating on single vertices and pairs of adjacent vertices where the meetings may take place.
The makespan for each suggestion for meeting-place(s) is computed in O(n) time (according to
the distances from the participants). Again, computing all-pairs-shortest-paths requires an overall
time of O(mn + n2 log n) using Johnson’s algorithm [CLRS01], which means that the total time
required for solving the problem is O(mn + n2 log n).
54
Cooperative TSP
3.3.3.2
Min-Makespan Sales cTSP
√
As noted before, the path version has an O( log n) approximation algorithm [KLS04], and there
is a lower bound of 5/3 on its approximation ratio (assuming P 6= N P ) [ABF+ 02]. The same
algorithm can clearly be used for the roundtrip version:
√
Claim 3.42. The roundtrip version of Min-Makespan Sales cTSP is O( log n)-approximable.
√
Proof. The known algorithm for the path version finds an O( log n) approximate solution
[KLS04]. Requiring that all the participants return to their original vertex at the end may increase
the cost of the solution found by the algorithm by a factor of at most 2. Clearly, the optimal
cost for the roundtrip version is at least the optimal cost for the path version. Therefore, this
√
problem also has an O( log n) approximation algorithm.
We now turn to providing a hardness of approximation result.
Claim 3.43. The roundtrip version of Min-Makespan Sales cTSP cannot be approximated
better than 5/4, unless P = N P .
Proof. We use a reduction from Set-Cover, similar to the reduction used in Theorem 3.35 for MinMax Full-Cooperation cTSP. We also use the same notations as in the proof of Theorem 3.35.
There are two differences in the construction of the reduction. First, we add another vertex w
which contains m agents (the number of sets) and is connected to all the vertices in VC . Second,
each of the vertices of VC contains a number of agents equal to its degree (which is the size of
the corresponding set). As in the above-mentioned reduction, all the edges have weight 1, the
vertices of VS contain one agent each, and v contains k − 1 agents.
A set-cover of size k (or less) provides a solution of cost 4 to our problem: The salesperson and
the agents in v visit the vertices in VC corresponding to the cover. The agents in these vertices
visit all the vertices in VS , and one of the agents who came from v visits w. Then the agents from
w visit all the non-visited vertices in VC . It is easy to verify that all the participants can return
from these visits without exceeding a makespan of 4.
On the other hand, assume there is a solution to the new problem with a makespan of 4. It
takes at least 2 time units to visit an agent in VS or w, so clearly these agents could not visit
other agents in VS , and the same is true for agents in VC which were first visited by agents from
w or Vs . Hence, agents in VS could either be visited by the salesperson, the k − 1 agents from v,
or agents in VC which these participants visited at the first time-unit. Since the salesperson and
3.4 Discussion and Open Problems
55
the agents from v could visit at most k vertices of VC in the first time unit, there is a set-cover of
size at most k.
Therefore, it is NP-hard to distinguish between an instance with minimum makespan of 5 and
an instance with minimum makespan of 4, which yields the required result.
3.3.3.3
Min-Makespan Full-Cooperation cTSP
Here we have tight upper and lower bounds of 2 on the approximation ratio. The upper bound for
the path version is obtained by simply letting all the agents go to the salesperson. Since delivering
the goods to the agent which is farthest from the salesperson takes at least half the travel-duration
between them, this yields a 2-approximation. The same can be done for the roundtrip version,
followed by a return of all the agents to their initial vertices. The optimum clearly requires here
at least the travel-duration to the furthest agent (at least the time until she receives the goods
plus the time for returning to her initial vertex if she moved). We thus have:
Corollary 3.44. Min-Makespan Full-Cooperation cTSP is 2-approximable for both the
path and roundtrip versions.
The hardness proofs for both the path and roundtrip versions are identical to the corresponding Min-Max problems (see Theorem 3.35 and Corollary 3.39). Therefore, we have:
Corollary 3.45. Both the path and the roundtrip versions of Min-Makespan Full-Cooperation cTSP cannot be approximated better than 2, assuming P6=NP.
3.4
Discussion and Open Problems
We obtained quite tight approximation and intractability results for most of the cTSP problems.
Some of the cTSP problems turn out to be easier (in sense of approximation) than the classical
TSP, while others are strictly harder.
√
The status of Min-Makespan Sales cTSP is not settled, as there is an O( log n) approxi-
mation and a constant inapproximability factor. Improving the factors of this problem as well as
tightening the factors for some others is yet to be achieved. It is also likely that the running time
of some of the PTAS can be improved.
There are some disturbing asymmetries in the Euclidean results (see Table 3.2). For example, while the roundtrip versions of Min-Sum Sales and Full-Cooperation cTSP have a
PTAS, the best approximations for the corresponding path-cTSP problems only guarantee some
constant factors. We conjecture that these two path-cTSP versions indeed have a PTAS, but we
56
Cooperative TSP
suspect that this may not be very easy to prove. This follows since it can be shown that a PTAS
for the first problem implies a (currently unknown) PTAS for the well-studied 3-bounded-degreeplanar-MST (e.g., [PV84, KRY96, FKK+ 97, Cha03, AC04]).
Chapter 4
On unweighted r-Gatherings
4.1
Introduction
Facility-location has been studied in many forms over the past decades (see, e.g., [AGK+ 04,
Byr07, Dre95, Gon85, GK99, GMM00, KM00, MYZ03, Svi02, Vyg05, ZCY04]). In the classic
metric facility-location problem, we are given a set of customer locations S and a set of potential
locations of facilities F (which may intersect S). Each location fi ∈ F is associated with a
cost p(fi ) for opening a facility there. For every si ∈ S and fj ∈ F , there is a cost d(si , fj )
for connecting a customer in si to a facility in fj . These costs are equivalent to distances, and
thus satisfy the symmetry and triangle-inequality requirements. The goal is to open facilities
and assign each customer to a facility, such that the total cost is minimized (i.e., the sum of the
facility opening-costs and the connection-costs should be minimal).
The metric facility-location problem models many realistic scenarios, in which service-posts
of a certain type should be opened to serve a set of customers. Applications range from classic
power-plants or warehouse location problems to locating servers in computer-networks (see, e.g.,
[Dre95, Vyg05] for surveys). The current best approximation algorithm for metric facility-location
achieves an approximation-ratio of 1.5 [Byr07]. On the other hand, this problem cannot be
approximated within a factor of less than 1.463, assuming P 6= N P [GK99].
One of the interesting recent variants of metric facility-location is the r-gathering problem,
introduced in parallel by Karger and Minkoff [KM00] and by Guha et al. [GMM00] (who called
it load-balanced facility-location). The basic additional requirement in the r-gathering problem is
that each facility will be assigned at least r customers (customers are not necessarily assigned to
the nearest open facility in this problem). This variant captures the idea that opening a facility is
economically justified only when it serves at least a certain amount of demand (and this constraint
58
may even be more natural than facility costs in some settings). Furthermore, in various settings
there is an inherent lower bound on the number of customers in each facility. For example, in
secret-sharing schemes (see [Sch96]), at least r shares are needed to uncover a secret. We may
need to locate servers in the network, to which clients will connect in order to uncover the secret,
and we may want this process to be as fast or as cheap as possible.
Both papers [GMM00, KM00] considered the generalization of r-gathering in which customers
have different demands, the connection-costs are the product of the demand and distance, and
each facility must serve customers having a total of at least r demand [GMM00, KM00]. They both
1+α
presented a ( 1−α
β, α) bicriteria approximation, for any α < 1, where β is the approximation-ratio
of the metric facility-location problem (currently 1.5 for the classic problem [Byr07] and 1.582
for the generalization in which customers may have different demands [Svi02]). Namely, their
algorithm guarantees that each open facility in the solution will serve at least αr demand, and the
cost will be at most
1+α
1−α β
times the optimal cost of the r-gathering problem. Choosing α =
r−1
r +
for the case of unit-demands provides a 1.5(2r − 1 + )-approximate feasible solution. Note that
we cannot hope for a significant improvement in the approximation-ratio due to improvement of
β, since β is lower-bounded by 1.463 [GK99]. Very recently, Svitkina [Svi08] obtained a constantfactor (single-criterion) approximation algorithm for this problem with unit-demands, providing
a 558-approximate solution.
Although the first papers considered minimizing the sum of costs [GMM00, KM00], a natural
variant is to minimize the maximal cost (in the spirit of the k-center problem [Gon85]). This may
model, for example, the time until all the facilities and connections will be available (if each cost
represents the time until the corresponding facility/connection will be ready). A special case of
the min-max version of this problem with unit-demands, called “r-gather clustering”, has been
recently considered by Aggrawal et al. [AFK+ 06]. In their special case, motivated by a clustering
application, all the facility costs are zero and all the locations of customers are included in the
set of optional facility locations (S ⊆ F ) [AFK+ 06]. Their paper presented a 2-approximation
algorithm for this case, and proved that it cannot be approximated better, for any r ≥ 7 (assuming
P 6= N P ). They also considered a generalization called (r, )-gather clustering, in which the
solution can ignore n of the customers (“outlier points”), and stated that this problem can be
approximated within a factor of 3 if facilities (cluster-centers) can only be located at customer
(input points) locations [AFK+ 06]. We note that unlike the algorithm of [GMM00, KM00], the
algorithm of [AFK+ 06] does not guarantee that each customer will be assigned to a nearest open
facility.
For the basic special case of r = 2, a recent paper of Anshelevich and Karagiozova [AK07]
proves that both min-sum 2-gathering without facility-costs and min-max 2-gathering can be
4.1 Introduction
59
solved in polynomial time.
Demaine et al. [DHM+ 07] have recently introduced another problem related to min-max 2gathering, which they called min-max minimum-movement facility location. In our terminology,
there are two types of customers in that problem: Customers from type A (“clients”) must be
assigned to a facility having at least one customer from type B (“server”) assigned to it, while
customers from type B do not have to be assigned. Also, S ⊆ F and there are no facility costs.
Demaine et al. [DHM+ 07] asked whether this problem can be approximated within a factor of
less than 2. We prove that the answer is negative, assuming P 6= N P .
In this chapter we focus on min-max r-gathering in the basic case of unit-demands - our results
refer to this problem unless stated otherwise. In addition to the basic r-gathering problem, we
consider the version in which there is an additional proximity requirement: Each customer in the
solution must be assigned to the nearest open facility. This is clearly a desirable property of a
solution in many facility-location settings, and also in clustering scenarios (e.g., in geographic
data-mining, see [GvKN06]). We manage to obtain a constant-factor approximation for this
problem as well.
4.1.1
Our results
We start by presenting a simple 3-approximation algorithm for min-max r-gathering. On the
other hand, we prove that this problem cannot be approximated within a factor of less than 3
(assuming P 6= N P ), for any r ≥ 3. By using a similar reduction, we also show that r-gather
clustering cannot be approximated within a factor of less than 2 for any r ≥ 3, thus improving
the hardness result of [AFK+ 06].
The same approximation algorithm extends to provide a 3-approximate solution for a generalization considered by [GMM00, KM00], in which each fi ∈ F has a different lower-bound ri on the
number of customers required. Furthermore, it extends to provide the same approximation-ratio
for the generalization in which there are several types of customers, and each open facility fi must
have at least rij customers of type j (this may be useful for example for achieving “p-Sensitive
k-Anonymity” [TB06] in publishing information from databases, similarly to the use of r-gather
clustering for achieving “k-Anonymity”[AFK+ 06]).
By using another extension of this algorithm, we provide a 3-approximation for the generalization of min-max r-gathering in which an -fraction of the customers can be ignored. We thus
match the approximation-ratio stated in [AFK+ 06] for the special case of (r, )-gather clustering.
Interestingly, practically the same algorithm also provides a 2r approximation for the minsum version of the problem, if there are no facility costs. For this case, this improves upon the
60
1.5(2r − 1) + approximation implied by the bicriteria algorithm of [GMM00, KM00]. In parallel
to our work, Lim et al. [LWX06] also obtained a 2r-approximation factor for this problem, using
a different algorithm.
Next we consider the proximity requirement and present a 9-approximation algorithm for
min-max r-gathering which satisfies it (i.e., each customer is assigned to a nearest open facility).
For the special case of r-gather clustering, our technique provides a 6-approximation algorithm.
In addition, we provide a 2-approximation algorithm for 2-gather clustering which satisfies the
proximity requirement. We show that this approximation factor cannot be improved: An algorithm for r-gather clustering which guarantees the proximity requirement cannot guarantee an
approximation-ratio smaller than 2.
Finally, we show that although min-max 2-gathering is polynomial [AK07], the related minmax minimum-movement facility-location [DHM+ 07] is NP-hard and cannot be approximated
within a factor of less than 2 (assuming P 6= N P ). This resolves the open-question recently posed
by Demaine et al. [DHM+ 07].
All our algorithms are based on discrete combinatorial techniques. Our hardness results use
reductions from Exact-k-cover and SAT.
The rest of this chapter is organized as follows. In Section 4.2 we present formal problem
definitions and notations. Section 4.3 presents our simple approximation algorithm for min-max
r-gathering, and analyzes its use for other versions. Section 4.4 considers the requirement of
assigning each customer to a nearest open facility. The hardness results are provided in Section
4.5. We end with some concluding remarks and open problems.
4.2
Problem Definitions and Notations
We now formally state the basic problems we consider and introduce some of the notations we
use. (We use slightly different notations from those of [GMM00, KM00].)
The input for an r-gathering problem consists of a set of customer-locations S = {s1 , ..., sn },
a set of potential facility-locations F = {f1 , ..., fm } with opening costs p : F → R+ ∪ {0}, and
distances (connection-costs) d : (S ∪ F ) × (S ∪ F ) → R+ ∪ {0}. The input also includes a positive
integer r > 1.
A solution is an assignment of the n customers to (not necessarily distinct) facilities, t1 , ..., tn ,
which are considered open, such that customer i is assigned to facility ti ∈ F , and the number of
customers assigned to each open facility is at least r. In the min-max version of the problem, the
goal is to minimize max1≤i≤n {max(d(si , ti ), p(ti ))} (we refer to this as the cost of the solution).
P
P
In the min-sum version, the goal is to minimize ni=1 d(si , ti ) + fi ∈{t1 ,...,tn } p(fi ) (each cost of
4.3 Approximating Min-Max r-Gathering
61
Algorithm Best-or-Rest
1. For each customer, find his min-cost, best facility and partners.
2. Sort the customers in non-decreasing min-cost order.
3. For each customer i in this sorted order:
If customer i and all his partners have not been assigned yet assign them to the best facility of customer i (open this facility
if it is not open yet). Otherwise, do nothing and continue to the
next customer.
4. Assign any unassigned customer to the nearest open facility. (In
case of a tie, arbitrarily choose the location with smallest index).
Figure 4.1: A 3-approximation algorithm for min-max r-gathering.
an open facility is considered once in this sum).
A special case of min-max r-gathering is r-gather clustering [AFK+ 06], where S ⊆ F , and
there are no facility costs (p(fi ) = 0, for 1 ≤ i ≤ m).
4.3
Approximating Min-Max r-Gathering
Definition 4.1. The “min-cost” of customer i, denoted c(i), is the minimum cost of assigning
r customers, including customer i, to a single facility (considering both the facility-cost and the
customers’ connection-costs). The location of this min-cost assignment, gi ∈ F , is called “the
best facility” of customer i. The “partners” of customer i are the r − 1 customers, other
than customer i, who participate in this min-cost assignment. (If there are several options we
arbitrarily prefer locations and customers with smaller indices).
We now provide a simple approximation algorithm for the problem, Best-or-Rest (see Figure
4.1).
Lemma 4.2. The cost of the solution found by algorithm Best-or-Rest for min-max r-gathering
is at most thrice the maximal min-cost.
Proof. First, observe that the cost of a customers’ assignment at stage (3) is the min-cost of one
of the customers assigned at this stage (customer i), which is at most the maximal min-cost of
62
any of the n customers.
Now consider a customer i assigned at stage (4). This customer was not assigned at stage
(3), which means that when customer i was considered at stage (3), at least one of his partners,
say customer j, had already been assigned to another facility, tj = gk (the best facility of some
customer k 6= i). Customer i can also be assigned to tj , with a cost of d(si , tj ). Clearly, d(si , tj ) ≤
d(si , sj ) + d(sj , tj ). Observe that d(si , sj ) ≤ 2c(i), since d(si , gi ) ≤ c(i) and d(gi , sj ) ≤ c(i) (as
j is one of the partners of customer i and gi is the best-facility of customer i). Also, d(sj , tj ) =
d(sj , gk ) ≤ c(k), since customer j is one of the partners of customer k. Since we performed stage
(3) in a non-decreasing order of min-cost, c(k) ≤ c(i). So taken together, for each customer
assigned at stage (4), d(si , ti ) ≤ 3c(i) (the customer is assigned to a nearest open facility, and we
saw that there exists an open facility which satisfies this). This yields the required result.
Theorem 4.3. Algorithm Best-or-Rest finds a 3-approximate solution for min-max r-gathering,
and can be implemented to run in O(n(m + r + log n)) time.
Proof. The cost of an optimal solution for the problem is clearly at least the maximal min-cost
(since there is a customer whose assignment requires at least that cost in any solution). Therefore,
the previous lemma proves that the algorithm finds a 3-approximate solution.
For implementing the first stage efficiently, we can first find for each t ∈ F the set of r
customers closest to t. This can easily be done in O(n) time for each facility (using selection). Let
Dt denote the distance from t to the r-th distant customer. Thus, for each customer i, the minimal
cost of assigning him along with r − 1 other customers to location t is max(Dt , d(si , t), p(t)). So
computing these costs for each customer and for each t ∈ F takes an overall time of O(mn). We
now find the best facility of each customer according to these costs (in an overall time of O(mn)).
The partners of customer i are clearly the r − 1 customers (other than customer i) that are closest
to his best facility. Given the sets of r closest customers that we computed for each facility, noting
the partners of each customer requires a total of O(rn) time (rn may be higher than mn). Thus,
stage (1) can be implemented to run in O(n(m + r)) time. Stage (2) clearly requires O(n log n)
time. The next stages are less time-consuming than the first one, and thus the total running time
is as stated.
Algorithm Best-or-Rest can also be used for the generalization in which an -fraction of the
customers may be ignored ( is specified in the input). We can simply ignore the n customers
with highest min-costs (in case of ties we ignore only those whose min-cost is strictly higher
than the min-cost of (1 − )n other customers), and then run this algorithm. This guarantees an
approximation-ratio of 3 for this generalization of the problem, since the optimal cost must be at
4.3 Approximating Min-Max r-Gathering
63
least the highest min-cost of the customers we considered (note that the customers we ignored
are not partners of customers we haven’t ignored, since their min-cost is higher). As mentioned
in the Introduction, this matches the approximation-ratio stated in [AFK+ 06] for a special case.
It is also easy to see that algorithm Best-or-Rest can be used to achieve the same approximation ratio even if there is a different lower-bound ri on the number of customers for each facility
fi ∈ F (a generalization considered by [GMM00, KM00] with the min-sum objective). This should
simply be taken into account in the definitions of min-cost, best-facility and partners, and the first
stage of the algorithm will change accordingly (and will be similarly implemented). Furthermore,
it can be used to achieve the same approximation-ratio for the generalization in which there are
several types of customers, and each open facility fi must have at least rij customers of type j
(again, this should simply be taken into account in Definition 4.1, changing the first stage of the
algorithm accordingly).
We next prove that algorithm Best-or-Rest can be used to provide a 2r approximation for
min-sum r-gathering (with unit demands), in the basic case introduced by [KM00] where there
are no facility costs. We call this case basic min-sum r-gathering. This improves upon the ratio
of 1.5(2r − 1) + implied by the algorithm of [GMM00, KM00] for this case of the problem. For
small values of r, it is also better than the recent 558 approximation of [Svi08].
We define the min-cost, best-facility and partners in the corresponding way for the min-sum
problem (the cost of an assignment to a facility is the sum of the connection-costs of the customers
rather than their maximum).
Lemma 4.4. The cost of the solution found by algorithm Best-or-Rest for basic min-sum rgathering is at most twice the sum of the min-costs of all the customers.
Proof. The proof is very similar to the proof of Lemma 4.2. First, observe that assigning customers
at a certain iteration of stage (3) costs exactly the min-cost of one of the customers being assigned
(customer i). which is clearly smaller than the sum of the min-costs of all the customers assigned.
Now consider a customer i assigned at stage (4). This customer was not assigned at stage
(3), which means that when customer i was considered at stage (3), at least one of his partners,
say customer j, had already been assigned to another facility, tj = gk (the best facility of some
customer k 6= i). Customer i can also be assigned to tj , with a cost of d(si , tj ). Clearly, d(si , tj ) ≤
d(si , sj ) + d(sj , tj ). Observe that d(si , sj ) ≤ c(i), since d(si , gi ) + d(gi , sj ) ≤ c(i) (as j is one
of the partners of customer i). Also, d(sj , tj ) = d(sj , gk ) ≤ c(k), since customer j was one of
the partners of customer k. Since we performed stage (3) in a non-decreasing order of min-cost,
c(k) ≤ c(i). So taken together, for each customer i assigned at stage (4), d(si , ti ) ≤ 2c(i) (since
the customer is assigned to the nearest open facility, and we saw that there exists a facility which
64
satisfies this). This yields the required result.
Lemma 4.5. The cost of an optimal solution for basic min-sum r-gathering is at least a (1/r)fraction of the sum of the min-costs of all the customers.
Proof. Consider a facility t ∈ F opened by an optimal solution OP T . Let x = yr + z be the
number of customers assigned to t in this solution (where y, z are integers such that y ≥ 1,
r > z ≥ 0). Now divide these customers into (y + 1) sets in the following way. For each customer
a assigned to t, calculate d(sa , t)/c(a). The first set, B0 , will contain the z customers for which
the above calculated value was maximal. The other customers are arbitrarily divided into y sets
of r customers, B1 , ..., By .
Consider a set Bi , 1 ≤ i ≤ y. For each customer a ∈ Bi , c(a) ≤
P
b∈Bi
d(sb , t) (since this is
the cost of assigning r customers, including customer a, to facility t). Summing this over all the
P
P
customers in Bi , we get a∈Bi c(a) ≤ r · a∈Bi d(sa , t). This is true for every 1 ≤ i ≤ y, which
means that the cost of assigning the customers of ∪yi=1 Bi in OP T is at least a (1/r)-fraction of
the sum of their min-costs.
Now consider a customer a ∈ B0 . If we replace one of the customers of B1 by customer a,
then the previous argument still holds for this modified set of r customers. So the total cost of
assigning the customers in this modified set to t is at least a (1/r)-fraction of the sum of their
min-costs. From the way B0 has been selected, it follows that d(sa , t)/c(a) ≥ 1/r (otherwise this
ratio must have been smaller than 1/r for all the customers in this set, and thus also for the
sums). Since this is true for any customer in B0 , it is true for the whole B0 , i.e., the cost of
assigning these customers to t is at least a (1/r)-fraction of the sum of their min-costs.
All the above is true for any facility t opened by an optimal solution, which means that the
cost of an optimal solution is at least a (1/r)-fraction of the sum of min-costs, as required.
Theorem 4.6. Algorithm Best-or-Rest finds a 2r-approximate solution for basic min-sum rgathering, and can be implemented to run in O(n(m + r + log n)) time for this problem.
Proof. The approximation-ratio follows from combining the last two lemmas. It is easy to see
that the running-time is the same as in Theorem 4.3, since we can similarly implement the first
stage of the algorithm (summing the costs in the min-cost computations instead of taking their
maximum), and the next stages are the same.
4.4
Assigning to a Nearest Open Facility
In this section we consider the min-max r-gathering problem with the additional constraint that
each customer should be assigned to the nearest open facility (or to one of the nearest open
4.4 Assigning to a Nearest Open Facility
65
Algorithm Move-to-Solid
1. Run algorithm Best-or-Rest. If there are no unsatisfied customers, we are done. Otherwise, reassign customers according
to the following stages (initially no customer is considered reassigned).
2. For each customer who has not been reassigned yet, check which
of the currently open facilities he prefers (in case of a tie choose
the facility with smallest index). If a facility is preferred by at
least r such customers, we say that it became solid.
3. Move to each solid facility all the customers who prefer it that
have not been reassigned yet. All the customers in solid facilities
are now considered reassigned (and will not be considered at the
next executions of stage (2)).
4. If there are non-solid facilities which contain less than r customers
now, reassign their remaining customers to the facilities they most
prefer out of the solid ones (and close these empty facilities).
5. If there are any non-solid facilities left, return to (2).
6. If there are unsatisfied customers, move them to the facilities they
prefer out of the remaining (solid) facilities.
Figure 4.2: A 9-approximation algorithm for min-max r-gathering, in which each customer
is assigned to a nearest open facility.
facilities in case of a tie). We start by presenting a 9-approximation algorithm which satisfies this
constraint.
In the following we say that a customer prefers a facility if there is no other open-facility
nearer to his input location. We use the term unsatisfied for a customer who is not assigned to a
nearest open facility. We use algorithm Move-to-Solid, described in Figure 4.2, for finding an
approximate solution.
Theorem 4.7. Algorithm Move-to-Solid finds a 9-approximate solution for min-max r-gathering,
in which each customer is assigned to a nearest open facility. It requires O(n3 /r + mn) time.
Proof. We first observe that the algorithm runs in the stated polynomial time. We call an ex-
66
ecution of stages (2)-(5) an iteration. Clearly, there are at most n/r open facilities after stage
(1), so there can be at most n/r iterations in which facilities become solid. Note that since there
are at least r customers in each facility after stage (1), there must be at least one solid facility.
If at a certain iteration no facility becomes solid, it means that at least one customer assigned
to a non-solid facility preferred one of the solid facilities at that iteration, and was therefore reassigned to it (the customers in non-solid facilities have not been reassigned yet, and if they all
prefer non-solid facilities in (2) then at least one of these facilities must be preferred by at least
r such customers). Since customers reassigned to solid facilities are not reassigned again until
stage (6), there can be at most n such iterations. Thus the number of iterations is smaller than
n + n/r. Clearly, each iteration requires O(n2 /r) time (this is what stage (2) may require at the
worst case). Stage (1) requires O(n(m + r + log n)) time according to Theorem 4.3, and stage
(6) can clearly be implemented in O(n2 /r) time. Summing these bounds yields the time bound
stated in the theorem (as r ≤ n).
We next explain why the algorithm indeed finds a solution for the problem. Since each solid
facility has at least r customers who preferred it over all the other remaining facilities, at least r
customers are left at each of the open facilities at the end (note that facilities are only closed and
not opened, so a cheaper assignment option cannot appear later). Since each customer is assigned
at stage (6) to a facility that he most prefers out of the remaining open facilities, each is assigned
to a nearest open facility (by definition). We thus turn to considering the cost.
We proved that algorithm Best-or-Rest finds a 3-approximate solution. We denote its cost
by C. We now prove that the reassignments of Move-to-Solid increase the cost of the solution
by a factor of at most 3. Note that the cost of open facilities does not increase (since we only close
facilities), so we only need to consider the increase in the customers’ connection-costs (distances).
Clearly, moving unsatisfied customers to a facility they prefer can only decrease their connectioncost. A customer’s connection-cost can increase only when he is moved from a canceled facility
(a facility found at stage (1) which was left with less than r customers) to the solid facility that
he most prefers (at stage (4)). Let u be such a canceled facility. If u was canceled, then one of
the customers assigned to it at stage (1) must have preferred one of the solid facilities at that
iteration, v, and was moved to it. Let customer i be the first such customer.
It is clear that d(u, v) ≤ 2C, since d(u, si ) ≤ C, and d(si , v) ≤ d(si , u) (since customer i
preferred v). Thus, moving any customer assigned to u at stage (1) to the solid facility that
he most prefers adds at most 2C to his connection-cost, which is therefore at most 3C. After
reaching a solid facility, the cost of a customer does not increase again (he is reassigned again
only if he is unsatisfied at the end, which may only decrease his cost). Therefore, the maximum
connection-cost of any customer in this solution is ≤ 3C, i.e., at most 9 times the optimum.
4.4 Assigning to a Nearest Open Facility
67
Figure 4.3: The instance constructed in the proof of Claim 4.8
We note that the procedure described in the last proof can be used to transform any solution
into a solution in which each customer is assigned to a nearest open facility, while increasing the
total cost by a factor of at most 3. Thus, by applying it to a 2-approximate solution found by
the algorithm of [AFK+ 06] for r-gather clustering, we can obtain a 6-approximate solution for
r-gather clustering which satisfies the proximity requirement. In the context of [AFK+ 06], it is a
clustering solution in which each object is assigned to a nearest cluster center (which is clearly a
desirable property of a clustering solution).
4.4.1
Improved Results for r = 2
Recall that min-max r-gathering is polynomial for r = 2 [AK07]. However, the solution found by
[AK07] does not necessarily satisfy the proximity requirement. We start by showing that for any
r ≥ 2, there are problem instances of r-gather clustering, for which the minimal cost solution that
satisfies the proximity requirement costs almost twice the optimum. We then provide algorithm
Nearest-Neighbor, that indeed finds a 2-approximate solution which satisfies the proximity
requirement for 2-gather clustering.
Claim 4.8. For every r ≥ 2 and > 0, there are instances of r-gather clustering such that the
minimum cost of a solution that satisfies the proximity requirement is at least (2 − ) times the
cost of an optimal solution that does not satisfy the requirement.
Proof. Consider a graph which is a simple path of four vertices: v1 , v2 , v3 , v4 . Assume that v1
contains r customers, v2 and v4 contain one customer each, and v3 contains r − 2 customers. Let
edge (v1 , v2 ) cost M , let the two other edges, (v2 , v3 ) and (v3 , v4 ), cost M + 1 each, and let the
connection-costs be those implied by the distances in this graph (all the vertices are potential
facility locations). This example is illustrated in Figure 4.3. The optimal solution costs M + 1:
The single customers are assigned to a facility in v3 , along with the r − 2 customers in that vertex
(and the customers in v1 are assigned to a facility in v1 ). However, the cheapest solution which
satisfies the proximity requirement is acheived when all the customers are assigned to a facility
in v3 , with a cost of 2M + 1. The required ratio follows, since M can be arbitrarily large.
68
Algorithm Nearest-Neighbor
1. For each customer i, find the customer j closest to him (his
nearest-neighbor), and let c(i) = d(si , sj ). (In case of a tie, pick
the customer with smallest index).
2. Consider the customers’ c(i) values in non-increasing order, and
do the following for each such value x:
(a) Build a graph G = (V, E), where V contains a vertex for each
customer i with c(i) = x that was not assigned yet. For every
u, v ∈ V , (u, v) ∈ E iff d(u, v) = x.
(b) Remove isolated vertices from G. Repeatedly remove edges
whose both endpoints have a degree > 1 as long as there are such
edges, i.e., until the graph becomes a set of vertex-disjoint stars.
(c) Open facilities in the star centers, and assign the customers
in the remaining vertices of G to their star’s center (in case of a
single edge, arbitrarily pick one of its endpoints to be the center)
(d) For each customer in V which was not assigned so far, open
a facility at the input location of his nearest-neighbor, and assign
that customer and his nearest neighbor to that facility.
Figure 4.4: Finding a 2-approximate solution for 2-gather clustering, in which each customer
is assigned to a nearest open facility.
Note that it is also easy to have an example in which si 6= sj if i 6= j. We can easily adjust
the above graph to comply with this requirement. Vertices v1 , v3 will be the centers of stars of r
or r − 2 other vertices, respectively, and will not contain a customer. Each of these new vertices
will contain one customer, and the edges of the stars will have a very small weight, 1 δ > 0. It
is easy to see again that the previous argument still holds for this modified graph, and the claim
follows.
For the approximation we use algorithm Nearest-Neighbor, described in Figure 4.4.
Theorem 4.9. Algorithm Nearest-Neighbor finds a 2-approximate solution for 2-gather clustering, in which each customer is assigned to a nearest open facility. It requires O(n2 ) time.
Proof. We start by showing that the algorithm finds a solution for the problem, which costs at
most maxi c(i). The cost of assigning a customer i at stage 2(c) is clearly at most c(i), since the
4.5 Hardness Results
69
assignment described uses at most one edge of E for each customer. Each open facility is assigned
at least two customers at this stage (those who are at the same star).
At stage 2(d), an unassigned customer i is assigned to the input location sj of his nearest
neighbor j. We observe that if customer j is the nearest neighbor of customer i then c(j) ≤ c(i)
(since customer i is at a distance of c(i) from customer j). If c(j) < c(i), then clearly customer
j was not assigned yet, and it is assigned to the same location sj by the algorithm (with zero
cost). So this is a valid assignment, and the cost of assigning customer i is exactly c(i). If
c(j) = c(i), then customer j must have been previously assigned to his own location sj , when
another customer, k (satisfying c(k) > c(j)), has been assigned to it (otherwise si would not have
been isolated in G, and customer i would have already been assigned at stage 2(c)). So this is
again a valid assignment, which costs c(i). Thus all the customers are assigned, and each facility
contains at least 2 customers.
All this is true for each of the c(i) values and for each of the customers. Therefore, the total
cost of the assignment is at most the maximum of the customers’ c(i) values. Clearly, the optimal
solution costs at least half of this (the customers might be able to meet at the middle of a shortest
path between them).
Finally, we explain why each customer is indeed assigned by the algorithm to a nearest open
facility. Facilities are only opened by the algorithm in locations of customers, and each customer
is either assigned to his own location or to the location of one of his nearest neighbors (in which
case there is no facility at his own location). As the algorithm progresses, there can only be
less assignment options (since some of the customers are already assigned to locations of other
customers). Therefore, at the end there can be no nearer open facility for any of the customers.
It is easy to see that each stage of the algorithm requires a total of at most O(n2 ) time.
4.5
Hardness Results
We match the approximation-ratio for min-max r-gathering with the following hardness result.
Theorem 4.10. For any r ≥ 3, it is NP-hard to approximate min-max r-gathering within a
factor of less than 3, even if there are no facility costs.
Proof. We prove the theorem by a reduction from the Exact-k-Cover problem (also called ExactCover by k-Sets), which is known to be strongly NP-hard for any k ≥ 3 [EKR99, GJ79]. The input
consists of a set of elements S = {x1 , ...xkn }, and m subsets of this set of elements, S1 , ..., Sm ,
where |Si | = k for every 1 ≤ i ≤ m. The question is whether there exists a collection of n subsets
Si1 , ..., Sin , such that each element is included in exactly one of them. Our reduction first proves
70
that min-max r-gathering is NP-hard, and we later see that this implies that it is NP-hard to
approximate within a factor of less than 3.
We construct the following input for min-max r-gathering. The set of customer locations
is S = {x1 , ...xkn }, i.e., there is one customer for each element xi , 1 ≤ i ≤ kn. There is one
potential facility location fi ∈ F for each subset Si (1 ≤ i ≤ m), with p(fi ) = 0. For every
xi ∈ Sj , d(xi , fj ) = 1. The other distances are those implied by this definition (i.e., the distances
in the graph G = (S ∪ F, E), where (u, v) ∈ E iff d(u, v) = 1 and the weight of each edge is 1).
We set r = k. We now prove that the cost of an optimal solution for this problem is 1 iff the
answer to the Exact-k-Cover problem is “yes”.
Assume the answer to the Exact-k-cover problem is “yes”. Opening facilities in the locations
corresponding to the cover subsets Si1 , ..., Sin , and assigning each customer to the facility corresponding to the subset which covers his corresponding element, provides a solution in which each
facility is assigned r customers and the cost is 1 for each customer. Thus, the optimal cost is
indeed 1.
On the other hand, if the optimal cost is 1, we show that the answer to the Exact-k-Cover
problem is “yes”. A solution with a cost of 1 can only exist if each customer is assigned to
a facility which corresponds to a subset containing his corresponding element. Thus, there are
exactly r such customers assigned to each open facility in that solution, since each facility has only
r customers at a distance of 1. Therefore there must be n such facilities, since all the customers are
assigned. These facilities correspond to n subsets, each of them containing r different elements.
Thus these subsets form an Exact-k-Cover. So both sides of the reduction are proven. Since
Exact-k-Cover is NP-hard for any k ≥ 3, we get that our problem is NP-hard for any r ≥ 3.
Clearly, the cost is at least 3 iff the answer is “no”, since there is no potential facility location
at distance 2 from a customer. Thus, the theorem is proven.
The problem remains hard to approximate even for the following special case.
Theorem 4.11. For any r ≥ 3, the special case of min-max r-gathering in which S = F and
there are no facility costs, is NP-hard to approximate within less than a factor of 2.
Proof. Proving NP-hardness for the special case where S=F requires a change in the reduction
described in the previous proof. Instead of having only one location fi corresponding to each
subset Si , we now have r locations, fi1 , ..., fir , corresponding to each subset Si . For each xj ∈ Si
we define d(xj , fi1 ) = 1. Also, for every 1 ≤ j < r, we define d(fij , fir ) = 1. An example can
be seen in Figure 4.5. Again, the other distances are those implied by those we defined. Each
location both contains a customer and is a potential location of a facility (S = F ). Again, r = k.
4.5 Hardness Results
71
Figure 4.5: An example illustrating the reduction in the proof of Theorem 4.11. The figure
shows the subgraph constructed for a subset Si = {x1 , x3 , x4 , x6 }
It is not difficult to see that a solution has cost 1 iff the customers corresponding to each subset
Si are assigned to fir , and customers who correspond to elements are assigned to neighboring
locations of type fi1 (as in the proof of the previous theroem, despite the additional locations
and customers). Otherwise the cost is at least 2. Therefore the reduction holds due to the same
arguments, and the problem cannot be approximated within a factor of less than 2, assuming
P 6= N P .
Since r-gather clustering is a generalization of the problem mentioned in the last theorem, this
hardness result also holds for r-gather clustering, thus matching the approximation-ratio obtained
by [AFK+ 06]. Previously this was known for r-gather clustering only for r ≥ 7 [AFK+ 06].
Corollary 4.12. For any r ≥ 3, it is NP-hard to approximate the r-gather clustering problem
within a factor of less than 2.
We next prove the hardness of a related problem described in the Introduction, min-max
minimum-movement facility-location, which was introduced by [DHM+ 07]. They observed that
this problem is approximable within a factor of 2. We provide a matching lower-bound on the
approximability, thus resolving an open question that they presented [DHM+ 07].
Theorem 4.13. It is NP-hard to approximate the min-max minimum-movement facility-location
problem within a factor of less than 2.
72
Figure 4.6: An example illustrating the reduction in the proof of Theorem 4.13. The letter
“c” represents a client and “s” represents a server.
Proof. The reduction is from SAT. We build an unweighted graph with the following vertices:
A “server” for each variable, a “client” for each clause, and an empty vertex for each literal,
connected to the clauses which contain it and to its variable (see Figure for an example illustrating
this construction). A facility may be located at any vertex. The connection-costs are defined
according to the distances in this graph. Thus, there is a satisfying assignment to the formula
iff there is a solution of cost 1 to the minimum-movement facility-location problem (facilities are
located in vertices corresponding to true literals). Otherwise, the cost is at least 2. Thus, the
theorem follows.
4.6
Concluding Remarks and Open Problems
We considered the min-max version of the r-gathering problem, and provided constant-approximation
algorithms and hardness-of-approximation results for several variants, some of which are tight.
Some of our results improve previous results for special cases or related problems, including an
improved approximation for min-sum r-gathering without facility costs and improved results for
4.6 Concluding Remarks and Open Problems
73
r-gather clustering and min-max minimum-movement facility-location.
Obvious remaining open problems are providing improved approximation algorithms or hardness results for min-max r-gathering with the proximity requirement and for min-sum r-gathering.
Other problems which remain for future research are the generalizations in which each customer may have a different demand and each facility must serve a total demand of at least
r, while the connection-costs are the product of distance and demand (previously considered by
[GMM00, KM00] for the min-sum version).
74
Chapter 5
Concluding Remarks
In this research we attempted to “explore new roads in an ancient land”, and show that meaningful
findings can still be obtained even without using “heavy tools”. As is often the case in theoretical
computer-science, it seems that the problems we introduced and the solutions we obtained raise
more open questions for future research than the number of open questions that we solved. We
hope that we succeeded in providing some new insights into the problems we considered, and that
future research will resolve the questions we left open, as well as other questions arising from
considering multicriteria, cooperative or other non-standard variants of combinatorial problems.
Contributing to the research in this field has been a great enriching experience, which I hope
many other prospective researchers will share.
76
Concluding Remarks
Bibliography
[AAS06]
A. Armon, A. Avidor, and O. Schwartz. Cooperative TSP. In Proceedings of the 14th
Annual European Symposium on Algorithms(ESA), Lecture Notes in ComputerScience, volume 4168, pages 41–50. Springer, 2006.
[ABF+ 02]
E. M. Arkin, M. A. Bender, S. P. Fekete, J. S. B. Mitchell, and M. Skutella. The
Freeze-Tag problem: How to wake up a swarm of robots. In Proceedings of the 13th
Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 568–577,
2002.
[ABG+ 03]
E. M. Arkin, M. A. Bender, D. Ge, S. He, and J. S. B. Mitchell. Improved approximation algorithms for the Freeze-Tag problem. In Proceedings of the 15th Annual
ACM Symposium on Parallel Algorithms and Architectures (SPAA), pages 295–303,
2003.
[AC04]
S. Arora and K. L. Chang. Approximation schemes for degree-restricted MST and
Red-Blue Separation problems. Algorithmica, 40(3):189–210, 2004.
[AFK+ 06]
G. Aggarwal, T. Feder, K. Kenthapadi, S. Khuller, R. Panigrahy, D. Thomas,
and A. Zhu.
Achieving anonymity via clustering.
In Proceedings of the 25th
ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems
(PODS), pages 153–162, 2006.
[AGK+ 04]
V. Arya, N. Garg, R. Khandekar, A. Meyerson, K. Munagala, and V. Pandit. Local
search heuristics for k-median and facility location problems. SIAM Journal on
Computing, 33(3):544–562, 2004.
[AH94]
E. Arkin and R. Hassin. Approximation algorithms for the geometric Covering
Salesman Problem. Discrete Applied Math., 55:197–218, 1994.
78
[AK07]
BIBLIOGRAPHY
E. Anshelevich and A. Karagiozova. Terminal backup, 3D matching, and covering
cubic graphs. In Proceedings of the 39th Annual ACM Symposium on Theory of
Computing (STOC), pages 391–400, 2007.
[Arc00]
A. Archer. Inapproximability of the asymmetric facility location and k-median
problems. citeseer.ist.psu.edu/archer00inapproximability.html, 2000.
[Arm08]
A. Armon. On min-max r-gatherings. In Approximation and Online Algorithms, 5th
International Workshop (WAOA 2007), Lecture Notes in Computer-Science, volume
4927, pages 128–141. Springer, 2008.
[Aro98]
S. Arora. Polynomial-time approximation schemes for Euclidean TSP and other
geometric problems. Journal of the ACM, 45(5):753–782, 1998.
[AZ06]
A. Armon and U. Zwick. Multicriteria global minimum cuts. Algorithmica, 46(1):15–
26, 2006.
[BEH00]
M. Bruglieri, M. Ehrgott, and H.W. Hamacher. Some complexity results for kcardinality minimum cut problems. Technical report. In Wirtschaftsmathematik,
University of Kaiserslautern, (69/2000), 2000.
[BME04]
M. Bruglieri, F. Maffioli, and M. Ehrgott. Cardinality constrained minimum cut
problems: Complexity and algorithms. Discrete Applied Mathematics, 137(3):311–
341, 2004.
[BNGNS98] A. Bar-Noy, S. Guha, J. Naor, and B. Schieber. Multicasting in heterogeneous networks. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing
(STOC), pages 448–453, 1998.
[Byr07]
J. Byrka. An optimal bifactor approximation algorithm for the metric uncapacitated
facility location problem. In Proceedings of the 10th International Workshop on
Approximation Algorithms for Combinatorial Optimization Problems (APPROX),
Lecture Notes in Computer Science, volume 4627, pages 29–43. Springer, 2007.
[Cha03]
T. M. Chan. Euclidean bounded-degree spanning tree ratios. In Proceedings of the
19th ACM Symposium on Computational Geometry (SoCG), pages 11–19, 2003.
[Chr76]
N. Christofides. Worst-case analysis of a new heuristic for the traveling salesman
problem. Technical report, Graduate School of Industrial Administration, Carnegy–
Mellon University, 1976.
BIBLIOGRAPHY
79
[Cli97]
J. Climacao. Multicriteria Analysis. Springer-Verlag, 1997.
[CLRS01]
T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduction to Algorithms. MIT Press, 2001.
[dBGK+ 05] M. de Berg, J. Gudmundsson, M. J. Katz, C. Levcopoulos, M. H. Overmars, and
A. F. van der Stappen. TSP with neighborhoods of varying size. Journal of Algorithms, 57:22–36, 2005.
[dBvKOS00] M. de Berg, M. van Kreveld, M. Overmars, and O. Schwarzkopf. Computational
Geometry: Algorithms and Applications. Springer, 2nd edition, 2000.
[DGKR05]
I. Dinur, V. Guruswami, S. Khot, and O. Regev. A new multilayered PCP and
the hardness of hypergraph vertex cover. SIAM Journal on Computing, 34(5):1129–
1146, 2005.
[DHM+ 07]
E. D. Demaine, M. Hajiaghayi, H. Mahini, A. S. Sayedi-Roshkhar, S. Oveisgharan,
and M. Zadimoghaddam. Minimizing movement. In Proceedings of the 18th Annual
ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 731–740, 2007.
[DM01]
A. Dumitrescu and J. S. B. Mitchell. Approximation algorithms for TSP with neighborhoods in the plane. In Proceedings of the 12th Annual ACM-SIAM Symposium
on Discrete Algorithms (SODA), pages 38–46, 2001.
[Dre95]
Z. Drezner. Facility Location. Springer-Verlag, 1995.
[EG00]
M. Ehrgott and X. Gandibleux. A survey and annotated bibliography of multicriteria combinatorial optimization. Operation Research Spectrum, 22:425–460, 2000.
[Ehr00]
M. Ehrgott. Multicriteria Optimization. Lecture Notes in Economics and Mathematical Systems. Springer-Verlag, 2000.
[EK01]
L. Engebretsen and M. Karpinski. Approximation hardness of TSP with bounded
metrics. In Proceedings of the 28th Annual International Colloquium on Automata,
Languages and Programming (ICALP), Lecture Notes in Computer-Science, volume
2076, pages 201–212. Springer, 2001.
[EKR99]
F. Ergun, R. Kumar, and R. Rubinfeld. Fast approximate PCPs. In Proceedings of
the 31st Annual ACM Symposium on Theory of Computing (STOC), pages 41–50,
1999.
80
[FKK+ 97]
BIBLIOGRAPHY
S. P. Fekete, S. Khuller, M. Klemmstein, B. Raghavachari, and N. Young. A network
flow technique for finding low-weight bounded-degree trees. Journal of Algorithms,
24:310–324, 1997.
[GGJ76]
M. R. Garey, R. L. Graham, and D. S. Johnson. Some NP-complete geometric problems. In Proceedings of the 8th Annual ACM Symposium on Theory of Computing
(STOC), pages 10–22, 1976.
[GH94]
O. Goldschmidt and Dorit S. Hochbaum. A polynomial algorithm for the k-cut
problem for fixed k. Mathematics of Operations Research, 19(1):24–37, 1994.
[GJ79]
M. R. Garey and D. S. Johnson. Computers and Intractability – A Guide to the
Theory of NP-Completeness. Freeman publishing, 1979.
[GK99]
S. Guha and S. Khuller. Greedy strikes back: Improved facility location algorithms.
Journal of Algorithms, 31(1):228–248, 1999.
[GL99]
J. Gudmundsson and C. Levcopoulos. A fast approximation algorithm for TSP with
neighborhoods. Nordic Journal of Computing, 6(4):469–488, 1999.
[GMM00]
S. Guha, A. Meyerson, and K. Munagala. Hierarchical placement and network
design problems. In Proceedings of the 41st Annual Symposium on Foundations of
Computer Science (FOCS), pages 603–612, 2000.
[Gon85]
T. F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Comput. Sci., 38:293–306, 1985.
[GT88]
A. V. Goldberg and R. E. Tarjan. A new approach to the maximum-flow problem.
Journal of the ACM, 35(4):921–940, 1988.
[GvKN06]
J. Gudmundsson, M. J. van Kreveld, and G. Narasimhan. Region-restricted clustering for geographic data mining. In Proceedings of the 14th Annual European Symposium on Algorithms (ESA), Lecture Notes in Computer-Science, volume 4168, pages
399–410. Springer, 2006.
[Han80]
P. Hansen. Bicriterion path problems. In G. Fandel and T. Gal, editors, Multiple
Criteria Decision Making: Theory and Applications, LNEMS 177, pages 109–127.
Springer-Verlag, Berlin, 1980.
BIBLIOGRAPHY
[HCP04]
81
S.P. Hong, S.J. Chung, and B.H. Park. A fully-polynomial bicriteria approximation
scheme for the constrained minimum spanning tree problem. Operations Research
Letters, 32(3):233–239, 2004.
[HHL88]
S. M. Hedetniemi, S. T. Hedetniemi, and A.L. Liestman. A survey of gossiping and
broadcasting in communication networks. Networks, 18(4):319–359, 1988.
[HO94]
J. Hao and J. B. Orlin. A faster algorithm for finding the minimum cut in a directed
graph. Journal of Algorithms, 17(3):424–446, 1994.
[Hoc82]
D. Hochbaum. Heuristics for the fixed cost median problem. Mathematical Programming, 22(1):148–162, 1982.
[Hoo91]
J. A. Hoogeveen. Analysis of Christofides’ heuristic: Some paths are more difficult
than cycles. Operation Research Letters, 10(5):291–295, 1991.
[HS76]
E. Horowitz and S. Sahni. Exact and approximate algorithms for scheduling nonidentical processors. Journal of the Association for Computing Machinery, 23:317–
327, 1976.
[JP99]
K. Jansen and L. Porkolab. Improved approximation schemes for scheduling unrelated parallel machines. In Proceedings of the 31st Annual ACM Symposium on
Theory of Computing (STOC), pages 408–417, 1999.
[Kar99]
D. R. Karger. Random sampling in cut, flow, and network design problems. Mathematics of Operations Research, 24(2):383–413, 1999.
[Kar00]
D. R. Karger. Minimum cuts in near-linear time. Journal of the ACM, 47(1):46–76,
2000.
[KLS04]
J. Konemann, A. Levin, and A. Sinha. Approximating the degree-bounded minimum
diameter spanning tree problem. Algorithmica, 41(2):117–129, 2004.
[KM00]
D. R. Karger and M. Minkoff. Building steiner trees with incomplete global knowledge. In Proceedings of the 41st Annual Symposium on Foundations of Computer
Science (FOCS), pages 613–623, 2000.
[KRY96]
S. Khuller, B. Raghavachari, and N. Young. Low degree spanning trees of small
weight. SIAM Journal on Computing, 25(2):355–368, 1996.
82
[KS96]
BIBLIOGRAPHY
D. R. Karger and C. Stein. A new approach to the minimum cut problem. Journal
of the ACM, 43(4):601–640, 1996.
[LST90]
J.K. Lenstra, D.B. Shmoys, and E. Tardos. Approximation algorithms for scheduling
unrelated parallel machines. Mathematical Programming, 46:259–271, 1990.
[LWX06]
A. Lim, F. Wang, and Z. Xu. A transportation problem with minimum quantity
commitment. Transportation Science, 40(1):117–129, 2006.
[Meg83]
N. Megiddo. Linear-time algorithms for linear programming in R3 and related
problems. SIAM Journal on Computing, 12(4):759–776, 1983.
[Mit99]
J. S. B. Mitchell. Guillotine Subdivisions approximate polygonal subdivisions: Part
II – A simple polynomial-time approximation scheme for geometric TSP, k-MST,
and related problems. SIAM Journal on Computing, 28(4):1298–1309, 1999.
[Mit07]
J. S. B. Mitchell. A PTAS for TSP with neighborhoods among fat regions in the
plane. In Proceedings of the 18th annual ACM-SIAM symposium on Discrete Algorithms (SODA), pages 11–18, 2007.
[MM95]
C. Mata and J. S. B. Mitchell. Approximation algorithms for geometric tour and
network design problems. In Proceedings of the 11th Annual ACM Symposium on
Computational Geometry (SoCG), pages 360–369, 1995.
[MS92]
C. L. Monma and S. Suri. Transitions in geometric minimum spanning trees. Discrete & Computational Geometry, 8:265–293, 1992.
[MYZ03]
M. Mahdian, Y. Ye, and J. Zhang. A 2-approximation algorithm for the softcapacitated facility location problem. In Proceedings of the 10th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), Lecture Notes in Computer Science, volume 2764, pages 129–140. Springer,
2003.
[NI92]
H. Nagamochi and T. Ibaraki. Computing edge-connectivity in multigraphs and
capacitated graphs. SIAM Journal on Discrete Mathematics, 5(1):54–66, 1992.
[NNI97]
H. Nagamochi, K. Nishimura, and T. Ibaraki. Computing all small cuts in an
undirected network. SIAM Journal of Discrete Mathematics, 10(3):469–481, 1997.
[Pap77]
C. H. Papadimitriou. Euclidean TSP is NP-complete. Theoretical Computer Science,
4:237–244, 1977.
BIBLIOGRAPHY
[PV84]
83
C. H. Papadimitriou and U. V. Vazirani. On two geometric problems related to the
Traveling Salesman Problem. Journal of Algorithms, 5:231–246, 1984.
[PY00]
C. H. Papadimitriou and M. Yannakakis. On the approximability of trade-offs and
optimal access of web sources. In Proceedings of the 38th Annual Symposium on
Foundations of Computer Science (FOCS), pages 86–92, 2000.
[Rav94]
R. Ravi. Rapid rumor ramification: Approximating the minimum broadcast time. In
Proceedings of the 35th Symposium on Foundations of Computer Science (FOCS),
pages 202–213, 1994.
[RZ00]
G. Robins and A. Zelikovsky. Improved Steiner tree approximation in graphs. In
Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms
(SODA), pages 770–779, 2000.
[SABM02]
M. O. Sztainberg, E. M. Arkin, M. A. Bender, and J. S. B. Mitchell. Analysis
of heuristics for the Freeze-Tag problem. In Proceedings of the 8th Scandinavian
Workshop on Algorithm Theory (SWAT), Lecture Notes in Computer-Science, volume 2368, pages 270–279. Springer, 2002.
[Sch96]
B. Schneier. Applied Cryptography, pages 71–73. John Wiley and Sons, 1996.
[SKK99]
K. Schloegel, G. Karypis, and V. Kumar. A new algorithm for multi-objective graph
partitioning. In Proceedings of the 5th European Conference on Parallel Processing (Euro-Par), Lecture Notes in Computer Science, volume 1685, pages 322–331.
Springer, 1999.
[SS05]
S. Safra and O. Schwartz. On the complexity of approximating TSP with Neighborhoods and related problems. Computational Complexity, 14:281–307, 2005.
[Svi02]
M. Sviridenko. An improved approximation algorithm for the metric uncapacitated
facility location problem. In Proceedings of the 9th International Conference on
Integer Programming and Combinatorial Optimization (IPCO), Lecture Notes in
Computer-Science, volume 2337, pages 240–257. Springer, 2002.
[Svi08]
Z. Svitkina. Lower-bounded facility location. In Proceedings of the 19th Annual
ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1154–1163, 2008.
[SW97]
M. Stoer and F. Wagner. A simple min-cut algorithm. Journal of the ACM,
44(4):585–591, 1997.
84
[TB06]
BIBLIOGRAPHY
T.M. Truta and V. Bindu. Privacy protection: p-sensitive k-anonymity property. In
Proceedings of the Workshop on Privacy Data Management, in conjunction with the
22nd IEEE International Conference of Data Engineering (ICDE), pages 94–103,
2006.
[Vaz03]
V. V. Vazirani. Approximation Algorithms. Springer-Verlag, 2003.
[Vyg05]
J. Vygen.
Approximation algorithms for facility location problems.
cite-
seer.ist.psu.edu/ vygen05approximation.html, 2005.
[Wel91]
E. Welzl. Smallest enclosing disks (balls and ellipsoids). In New Results and New
Trends in Computer Science, Lecture Notes in Computer Science, volume 555, pages
359–370. Springer, 1991.
[ZCY04]
J. Zhang, B. Chen, and Y. Ye. Multi-exchange local search algorithm for the capacitated facility location problem. In Proceedings of the 10th International Conference
on Integer Programming and Combinatorial Optimization (IPCO), Lecture Notes in
Computer-Science, volume 3064, pages 219–233. Springer, 2004.
‫אוניברסיטת תל אביב‬
‫הפקולטה למדעים מדויקים ע"ש ריימונד ובברלי סאקלר‬
‫ביה"ס למדעי המחשב ע"ש בלבטניק‬
‫אלגוריתמים מקורבים ומדויקים לפתרון‬
‫גרסאות חדשות של בעיות קלאסיות‬
‫בגרפים‬
‫חיבור זה הוגש לשם קבלת תואר דוקטור לפילוסופיה ע"י‬
‫אמתי ערמון‬
‫בהנחיית פרופ' אורי צוויק‬
‫הוגש לסנאט של אוניברסיטת תל אביב‬
‫ספטמבר ‪2008‬‬
‫תשרי התשס"ט‬
‫מוקדש לאימי ולסבתי בלינה‬
‫תודות‬
‫למנחה שלי‪ ,‬אורי צוויק‪ ,‬על ההנחיה המוצלחת של המחקר‪ ,‬ועל כך שחלק עימי את גישתו האופטימית‬
‫לפתרון בעיות אלגוריתמיות‪.‬‬
‫לעמוס פיאט‪ ,‬מיכה שריר‪ ,‬ורה אסודי‪ ,‬אייל אבן‪-‬דר ובן זנדבנק‪ ,‬על שיתופי‪-‬הפעולה הממושכים‬
‫והמהנים בהוראת קורסים לתואר ראשון‪.‬‬
‫לעדי אבידור ועודד שוורץ‪ ,‬על שיתוף‪-‬פעולה מחקרי פורה ועבודה משותפת מהנה‪.‬‬
‫לאמיר אפשטיין ועידו צמרת‪ ,‬שנהניתי לחלוק איתם משרד‪.‬‬
‫ולאימי ולסבתי בלינה‪ ,‬על כל האהבה והתמיכה‪ ,‬ועל כך שעשו אותי מי שאני‪.‬‬
‫תמצית‬
‫גרפים הם ככל הנראה האובייקט הנחקר ביותר בתיאוריה של מדעי המחשב‪ ,‬ובעיות רבות הקשורות‬
‫אליהם נחקרות מזה עשורים רבים‪ ,‬כדוגמת מציאת מסלולים קצרים ביותר‪ ,‬זרימה מקסימלית‪,‬‬
‫ובעיית הסוכן הנוסע‪ .‬עבור חלק מהבעיות בגרפים פותחו אלגוריתמים יעילים‪ ,‬בעוד שאחרות הוכחו‬
‫כקשות לפתרון‪ .‬עבור רבות מהבעיות הקשות פותחו אלגוריתמים יעילים שמוצאים פתרון מקורב‬
‫לבעיה‪ ,‬כלומר פתרון שקרוב להיות אופטימלי‪.‬‬
‫בעבודה זו אנו עוסקים בגרסאות חדשות של שלוש בעיות מרכזיות בגרפים‪ :‬מציאת חתך מינימלי‬
‫גלובלי‪ ,‬בעיית הסוכן הנוסע‪ ,‬ובעיית מיקום המתקנים )‪ .(facility location‬עבור כל אחת מהבעיות‬
‫האלה‪ ,‬הגרסאות בהן אנו עוסקים מכלילות או מרחיבות את הבעיה‪ ,‬ע"י הוספת מטרות‪ ,‬אילוצים‪ ,‬או‬
‫אפשרויות‪ ,‬הנובעים מתרחישים מציאותיים או ממחקר תיאורטי קודם‪ .‬במחקרנו על הגרסאות הללו‪,‬‬
‫אנו עונים גם על מספר שאלות פתוחות שהוצגו לגבי מקרים פרטיים שנחקרו בעבר‪ ,‬ואנו משפרים‬
‫מספר תוצאות קודמות‪.‬‬
‫בפרק ‪ 2‬אנו עוסקים בגרסאות מרובות‪-‬קריטריונים של בעיית החתך המינימלי הגלובלי‪ .‬בהכללת‬
‫הבעיה הזו ל‪ k-‬קריטריונים‪ ,‬לכל קשת יש ‪ k‬מחירים אי‪-‬שליליים‪ .‬המחירים הללו נמדדים ביחידות‬
‫שונות‪ ,‬שלא ניתן להמיר ביניהן‪ .‬בגרסת ה‪"-‬וגם" של הבעיה‪ ,‬בחירת קשת דורשת תשלום של כל ‪k‬‬
‫המחירים שלה‪ .‬בגרסת ה‪"-‬או"‪ ,‬ניתן לבחור איזה מבין המחירים שלה לשלם‪ .‬בהינתן ‪ k‬חסמים‪ ,‬אחד‬
‫בכל קריטריון‪ ,‬בעיית ההחלטה הבסיסית היא‪ :‬האם קיים חתך שהעלות של בחירת קשתותיו אינה‬
‫חורגת מאף אחד מהחסמים‪.‬‬
‫אנו מראים שגרסת ה‪"-‬וגם" ניתנת לפתרון בזמן פולינומיאלי לכל מספר קבוע של קריטריונים )שאינו‬
‫חלק מהקלט(‪ ,‬והיא ‪-NP‬קשה עבור מספר קריטריונים לא קבוע‪ .‬התוצאות שלנו עשויות להיראות‬
‫מפתיעות‪ ,‬כיוון ש‪ Papadimitriou and Yannakakis [PY00] -‬הוכיחו שבעיית ה‪ s-t -‬חתך המינימלי‬
‫עם שני קריטריונים היא ‪-NP‬קשה‪ .‬העבודה שלנו פותרת שאלה פתוחה שהציגו ‪Bruglieri et al.‬‬
‫]‪ .[BEH00, BME04‬הם שאלו האם מגבלה על מספר קשתות החתך הופכת את בעיית החתך‬
‫המינימלי הגלובלי לבעיה ‪-NP‬קשה או שהיא עדיין פולינומיאלית‪ .‬אנו מראים שזהו מקרה פרטי של‬
‫בעיית ה‪"-‬וגם" שחקרנו‪ ,‬שהוא פולינומיאלי‪.‬‬
‫לגבי גרסת ה‪"-‬או"‪ ,‬אנו מראים ‪-NP‬קשיות אפילו עבור שני קריטריונים‪ ,‬ומוכיחים שהבעיה ניתנת‬
‫לפתרון בזמן פסאודו‪-‬פולינומיאלי לכל מספר קבוע של קריטריונים )שאינו חלק מהקלט(‪ .‬אנו מראים‬
‫שלבעיה הזו יש גם ‪ .FPTAS‬בהמשך אנו עוסקים בהרחבות וישומים‪ ,‬וכן בגרסת "או" מרובת‬
‫קריטריונים של שתי בעיות אופטמיזיציה נוספות‪ .‬למיטב ידיעתנו‪ ,‬אנו הראשונים שעוסקים בגרסאות‬
‫"או" של בעיות מרובות קריטריונים‪ .‬פרק זה מבוסס על המאמר ]‪.[AZ06‬‬
‫בפרק ‪ 3‬אנו עוסקים בגרסאות שיתופיות של בעיית הסוכן הנוסע )‪ .(TSP‬בבעיות האלה סוכן‪-‬נוסע‬
‫צריך להעביר סחורה ללקוחות שמוכנים לסייע לתהליך ההפצה‪ .‬המוטיבציה הבסיסית לעיסוק‬
‫בגרסאות כאלה היא שבמקרים מציאותיים רבים ה‪"-‬לקוחות" הם למעשה אנשים ששייכים לאותו‬
‫ארגון‪/‬חברה שהסוכן הנוסע שייך אליו‪ ,‬כך שאפשר לבקש מהם לעזור‪ .‬הלקוחות עשויים להיות‬
‫יכולים לשתף‪-‬פעולה בכמה אופנים‪ :‬על‪-‬ידי תזוזה לכיוון הסוכן הנוסע כדי לקבל את הסחורה‪ ,‬על‪-‬ידי‬
‫העברת הסחורה שהם קיבלו ללקוחות אחרים‪ ,‬או על‪-‬ידי עשיית שני הדברים האלה‪ .‬יש כמה‬
‫פונקציות‪-‬מטרה שעשויות להיות רלוונטיות במקרים כאלה‪ :‬מינימיזציה של סכום המרחקים שנסעו‬
‫כל המשתתפים‪ ,‬מינימיזציה של המרחק המקסימלי שנסע איזשהו משתתף‪ ,‬או מינימיזציה של הזמן‬
‫עד שכל תהליך ההפצה יסתיים‪.‬‬
‫המחקר שלנו עוסק בכל הקומבינציות של אופני שיתוף‪-‬הפעולה ופונקציות‪-‬המטרה‪ ,‬בגרפים לא‪-‬‬
‫מכוונים ממושקלים‪ ,‬וגם במרחב אוקלידי ממימד קבוע‪ .‬אנו מראים שלרוב הבעיות האלה יש‬
‫אלגוריתם קירוב שמשיג יחס קבוע‪ ,‬לרבות מהאחרות יש ‪ ,PTAS‬וכמה מהבעיות ניתנות לפתרון בזמן‬
‫פולינומיאלי‪ .‬בצד הקשיות‪ ,‬אנו נותנים הוכחות ‪-NP‬קשיות והוכחות של קשיות קירוב עבור קבועים‬
‫מסויימים‪ ,‬שחלקן הדוקות‪ .‬כל האלגוריתמים שלנו הם קומבינטוריים לחלוטין‪ ,‬והוכחות הקשיות‬
‫שלנו משתמשות ברדוקציות מבעיות ‪-NP‬קשות ידועות‪ ,‬ללא צורך בשימוש במשפט ה‪ .PCP -‬פרק זה‬
‫מבוסס על המאמר ]‪.[AAS06‬‬
‫בפרק ‪ 4‬אנו עוסקים בגרסת ‪ min-max‬של בעיית ‪ ,r-gathering‬שנחקרה בעבר‪ ,‬עם משקלי‪-‬יחידה‪.‬‬
‫הבעיה שאנו עוסקים בה היא בעיית מיקום‪-‬מתקנים מטרית‪ ,‬שבה כל מתקן פתוח חייב לשרת לפחות‬
‫‪ r‬לקוחות‪ ,‬והמקסימום של מחירי המתקנים והחיבור צריך להיות מינימלי )במקום הסכום(‪.‬‬
‫המוטיבציה לבעיה הזו היא תרחישים שבהם דרושים ‪ r‬לקוחות כדי שיהיה כדאי לפתוח מתקן‪,‬‬
‫והמחירים מייצגים את הזמן עד שהמתקן‪/‬החיבור יהיה מוכן )כלומר אנו רוצים שהפתרון כולו יהיה‬
‫מוכן מוקדם ככל האפשר(‪.‬‬
‫אנו מציגים אלגוריתם שמשיג יחס קירוב ‪ 3‬עבור הבעיה הזו‪ ,‬ומוכיחים שהשגת קירוב טוב יותר היא‬
‫‪-NP‬קשה‪ .‬לאחר מכן אנו עוסקים בבעיה הזו עם האילוץ הטבעי הנוסף‪ ,‬לפיו כל לקוח יחובר למתקן‬
‫הפתוח הקרוב אליו ביותר‪ .‬עבור גרסה זו אנו מציגים אלגוריתם שמשיג יחס קירוב ‪ .9‬בנוסף אנו‬
‫עוסקים בגרסאות ומקרים פרטיים שנחקרו בעבר‪ ,‬ומשיגים תוצאות אלגוריתמיות ותוצאות קשיות‬
‫משופרות‪ .‬תוצאות הפרק הזה מבוססות על המאמר ]‪.[Arm08‬‬
‫תקציר‬
‫גרפים הם ככל הניראה האובייקט הנחקר ביותר בחקר התאוריה של מדעי‪-‬המחשב‪ ,‬ובעיות רבות‬
‫בגרפים נחקרו במשך עשורים רבים‪ .‬רשימה חלקית של בעיות כאלה כוללת את בעיית מציאת‬
‫המסלולים הקצרים ביותר‪ ,‬עץ‪-‬פורש מינימלי‪ ,‬זרימה מקסימלית‪ ,‬חתך מינימלי גלובלי‪ ,‬הסוכן הנוסע‪,‬‬
‫עץ שטיינר מינימלי‪-k ,‬מרכז‪-k ,‬חציון‪ ,‬ועוד‪ .‬ניתן למצוא סקירה בסיסית בספר היסודי של ‪Cormen‬‬
‫]‪ .et al. [CLRS01‬עבור חלק מהבעיות הקלאסיות הללו פותחו אלגוריתמים יעילים‪ ,‬בעוד שאחרות‬
‫הוכחו כקשות לפתרון‪ .‬עבור רבות מהבעיות הקשות פותחו אלגוריתמים יעילים שמוצאים פתרון‬
‫מקורב )ראו למשל סקירה בספר ]‪.([Vaz03‬‬
‫במקביל למחקר המתמשך של הבעיות הקלאסיות בגרפים‪ ,‬יש מחקרים רבים על מקרים פרטיים‬
‫שלהן‪ ,‬הכללות‪ ,‬וגרסאות אחרות‪ .‬מקרים פרטיים עוסקים פעמים רבות בסוגים מיוחדים של גרפים‪,‬‬
‫או במקרים שבהם יש מגבלות על ערכי קלט מסויימים )כמו הגבלתם להיות שלמים‪ ,‬או הגבלה לטווח‬
‫מספרים מסויים(‪ .‬הכללות עוסקות‪ ,‬למשל‪ ,‬בגרפים מכוונים במקום גרפים לא מכוונים‪ ,‬או‬
‫במטרואידים‪ .‬סוגים אחרים של גרסאות מערבים שינוי של פונקציית המטרה )סכום‪/‬מקסימום‪/‬‬
‫מינימום‪/‬וכו'(‪ ,‬תוספת‪/‬הסרה של אילוץ‪ ,‬ו‪/‬או התייחסות למרחב שונה )למשל למישור או למרחב‬
‫אוקלידי אחר(‪ .‬המוטיבציה לגרסאות השונות נבעה לפעמים משאלות תאורטיות‪ ,‬ולפעמים מישומים‬
‫מציאותיים‪.‬‬
‫בעבודה זו אנו עוסקים בגרסאות חדשות של שלוש בעיות קלאסיות‪ :‬החתך המינימלי‪ ,‬הסוכן הנוסע‪,‬‬
‫ומיקום מתקנים‪ .‬עבור כל אחת מהבעיות‪ ,‬ניראה שהגרסה שבה אנו עוסקים מאירה על הבעיה באור‬
‫חדש‪ ,‬ומכלילה גרסאות שנחקרו בעבר‪ .‬עבודתנו עונה על מספר שאלות פתוחות שנשאלו לגבי‬
‫הגרסאות האלה‪ ,‬משפרת מספר תוצאות קודמות‪ ,‬ומרחיבה את היריעה של המחקר הקודם‪.‬‬
‫חתכים מינימליים מרובי‪-‬קריטריונים‬
‫בפרק ‪ 2‬אנו עוסקים בגרסאות מרובות קריטריונים של בעיית החתך המינימלי הגלובלי‪ .‬בבעיה הזו יש‬
‫לחלק לשתי קבוצות לא‪-‬ריקות את הקודקודים של גרף לא‪-‬מכוון שקשתותיו ממושקלות‪ .‬המטרה‬
‫היא שסכום המשקלים )מחירים( של הקשתות המקשרות בין שתי תתי‪-‬הקבוצות יהיה מינימלי‪.‬‬
‫‪ Karger‬הציג אלגוריתם מונטה‪-‬קרלו כמעט לינארי עבור הבעיה הזאת‪ ,‬שרץ בזמן )‪O(mlog3n‬‬
‫]‪ .[Kar00‬האלגוריתמים הדטרמיניסטיים המהירים ביותר עבור הבעיה הזו דורשים זמן של‬
‫)‪ .[HO94, NI92, SW97] O(mnlogn‬זה זמן דומה לזה שדורשים האלגוריתמים המהירים ביותר‬
‫למציאת חתך ‪ s-t‬מינימלי‪ ,‬חתך שמפריד בין שני קודקודים מסויימים‪ s ,‬ו‪ ,t-‬שהם חלק מקלט‬
‫הבעיה )הבעיה הזו נפתרת על‪-‬ידי אלגוריתמים למציאת זרימה מקסימלית‪ ,‬ראו למשל ]‪.([GT88‬‬
‫בעבודות של ‪ [BEH00, BME04] Bruglieri et al.‬הוצגה הבעיה של מציאת חתך‪-‬מינימלי "בעל‬
‫עוצמה מוגבלת"‪ ,‬כלומר חתך מינימלי במשקלו מבין החתכים שבהם מספר קשתות‪-‬החתך הוא קטן‬
‫מערך מסויים שמופיע בקלט לבעיה‪ .‬המחברים שאלו האם הבעיה הזו ניתנת לפתרון בזמן‬
‫פולינומיאלי‪ ,‬והשאירו את השאלה הזו כשאלה פתוחה‪ .‬ניתן לראות את הבעיה שהם הציגו כמקרה‬
‫פרטי של גרסה דו‪-‬קריטריונית של בעיית החתך המינימלי הגלובלי‪ ,‬שבה לכל קשת יש מחיר נוסף‬
‫)ששווה ל‪ 1-‬עבור כל הקשתות(‪ ,‬והמינימיזציה עבור המחיר )קריטריון( הראשון צריכה להיעשות מבלי‬
‫לחרוג מגבול מסויים עבור המחיר השני )בהנחה שלא ניתן להמיר בין שני המחירים(‪.‬‬
‫יש עבודות רבות‪ ,‬בעיקר בקהילת חקר‪-‬הביצועים‪ ,‬על בעיות עם עוצמה מוגבלת ובעיות מרובות‪-‬‬
‫קריטריונים‪ ,‬שהן בדרך כלל ‪-NP‬קשות ודורשות קירובים )ראו למשל ]‪.([EG00, Ehr00, Cli97‬‬
‫תוצאה ספציפית שקשורה לבעיה הזו היא תוצאה של ]‪,Papadimitriou and Yannakakis [PY00‬‬
‫שהוכיחו שהגרסה הדו‪-‬קריטריונית של מציאת חתך ‪ s-t‬מינימלי היא ‪-NP‬קשה‪.‬‬
‫התרומה שלנו‬
‫באופן די מפתיע‪ ,‬אנו מראים שהגרסה הדו‪-‬קריטריונית של חתך מינימלי גלובלי ניתנת לפתרון בזמן‬
‫פולינומיאלי‪ .‬בכך אנו פותרים גם את השאלה הפתוחה שהציגו ‪ .Bruglieri et al.‬אנו מרחיבים את‬
‫המחקר שלנו ועוסקים בשתי גרסאות מרובות‪-‬קריטריונים כלליות יותר של בעיית החתך המינימלי‬
‫הגלובלי‪ .‬בתרחיש ה‪-k -‬קריטריוני‪ ,‬לכל קשת יש ‪ k‬מחירים )אי‪-‬שליליים(‪ .‬המחירים הללו נמדדים‬
‫ביחידות שונות‪ ,‬שלא ניתן להמיר ביניהן‪ .‬בגרסת ה‪"-‬וגם" של הבעיה‪ ,‬הכללת קשת בחתך דורשת‬
‫לשלם את כל המחירים שלה‪ .‬בגרסת ה‪"-‬או" של הבעיה‪ ,‬ניתן לבחור את אחד ממחירי הקשת‪ ,‬ורק‬
‫הוא ישולם )ביחידות המתאימות(‪ .‬בהינתן ‪ k‬חסמים‪ ,‬השאלה הבסיסית היא האם יש חתך של הגרף‬
‫שמחירו לא חורג באף קריטריון מהחסמים הנתונים‪.‬‬
‫אנו מראים שגרסת ה‪"-‬וגם" של הבעיה היא פולינומיאלית לכל מספר קבוע של קריטריונים‪ ,‬והיא‬
‫‪-NP‬קשה עבור מספר קריטריונים לא קבוע )כלומר מספר שהוא חלק מהקלט(‪ .‬לעומת זאת‪ ,‬גרסת ה‪-‬‬
‫"או" של הבעיה היא ‪-NP‬קשה אפילו עבור ‪ ,k=2‬אבל ניתנת לפתרון פסאודו‪-‬פולינומיאלי לכל מספר‬
‫קבוע של קריטריונים‪ .‬לבעיה הזו יש גם ‪ .FPTAS‬אנו מציגים תוצאות דומות לגרסאות‬
‫האופטימיזציה של שתי הבעיות האלה )מינימיזציה של המחיר בקריטריון מסויים‪ ,‬בלי לחרוג‬
‫מהחסמים בקריטריונים האחרים(‪ .‬בנוסף אנו עוסקים גם בהרחבות וישומים נוספים‪ ,‬כולל גרסאות‬
‫מרובות‪-‬קריטריונים מסוג "או" של שתי בעיות אופטימיזציה נוספות‪ .‬למיטב ידיעתנו עבודה זו היא‬
‫הראשונה לעסוק בגרסאות "או" של בעיות מרובות‪-‬קריטריונים‪ .‬הפרק הזה מבוסס על המאמר‬
‫]‪.[AZ06‬‬
‫בעיות סוכן‪-‬נוסע עם שיתוף‬
‫בפרק ‪ 3‬אנו עוסקים בגרסאות של בעיית הסוכן‪-‬הנוסע שיש בהן שיתוף בין הסוכן לבין הלקוחות‪.‬‬
‫הקלט לבעיית הסוכן הנוסע הקלאסית כולל גרף מלא לא‪-‬מכוון עם משקלים על הקשתות‪ ,‬והמטרה‬
‫היא למצוא מעגל פשוט שעובר בין כל צמתי הגרף שמשקלו הכולל מינימלי‪ .‬המוטיבציה הקלאסית‬
‫לבעיה הזו היא תכנון מסלול‪-‬נסיעה עבור סוכן‪-‬נוסע‪ ,‬שצריך לבקר אוסף של ערים‪/‬לקוחות ולשוב‬
‫לביתו‪ ,‬כך שסך מרחק הנסיעה יהיה מינימלי‪ .‬הבעיה הזו ידועה כבעיה ‪-NP‬קשה ]‪ ,[GGJ76‬ועבור‬
‫משקלי‪-‬קשתות כלליים לא ניתן להשיג לה אפילו יחס קירוב שהוא פולינומיאלי בגודל הקלט‬
‫]‪ .[Vaz03‬עבור הגרסה המטרית‪ ,‬שבה משקלי הקשתות מקיימים סימטריה ואי‪-‬שיויון המשולש‪,‬‬
‫‪ Christofides‬הציג אלגוריתם קירוב שמשיג יחס ‪ .[Chr76] 3/2‬עבור הגרסה הזו הוכח שלא ניתן‬
‫להשיג יחס קירוב טוב יותר מאשר ‪) 131/130‬בהנחה ש‪ P-‬שונה מ‪.[EK01] (NP -‬‬
‫לעומת זאת‪ ,‬בכל מרחב אוקלידי ממימד קבוע‪ ,‬יש לבעיה הזאת ‪ ,[Aro98, Mit99] PTAS‬והיא עדיין‬
‫‪-NP‬קשה ]‪ .[Pap77‬הוצג גם אלגוריתם קירוב שמשיג יחס ‪ 3/2‬לגרסת המסלול של הבעיה ]‪,[Hoo91‬‬
‫כלומר הגרסה שבה הסוכן לא צריך לשוב לנקודת ההתחלה שלו )צריך למצוא מסלול ולא מעגל‪,‬‬
‫כשנקודת קצה אחת של המסלול נתונה בקלט(‪.‬‬
‫בעיית הסוכן‪-‬הנוסע נחקרה במשך השנים בגרסאות רבות‪ .‬אחת מהגרסאות המעניינות האחרונות‬
‫היא בעיה שמכונה בעיית ‪ ,[ABF+02, SABM02, ABG+03,KLS04] Freeze-Tag‬שהוצגה על‪-‬ידי‬
‫‪ .[ABF+02] Arkin et al.‬המוטיבציה לבעיה הזו נובעת מתרחיש שקיים בהפעלת בנחילי רובוטים‪.‬‬
‫בתרחיש הזה יש רובוט מופעל אחד שצריך להפעיל אוסף של רובוטים אחרים הפזורים בשטח‪ .‬כדי‬
‫להפעיל רובוט יש להגיע אליו‪ ,‬וכשרובוט מופעל ניתן להעביר לו הנחיות תנועה כך שהוא יפעיל‬
‫רובוטים אחרים‪ .‬המטרה היא להשלים את הפעלת כל הרובוטים מהר ככל האפשר‪.‬‬
‫‪ Arkin et al.‬הציגו אלגוריתמים שמשיגים יחס קירוב קבוע עבור הבעיה הזאת עבור כמה סוגים‬
‫מיוחדים של גרפים‪ ,‬והוכיחו שלא ניתן להשיג יחס קירוב טוב יותר מ‪ 5/3 -‬עבור הבעיה )בהנחת ‪P‬‬
‫שונה מ‪ Konemann et al. .(NP -‬הציגו לאחר מכן אלגוריתם שמשיג קירוב )‪ O( log n‬בגרף כללי‬
‫]‪.[KLS04‬‬
‫ניתן לראות את בעיית ‪ Freeze-Tag‬כגרסה של בעיית הסוכן הנוסע‪ ,‬שבה ה‪"-‬לקוחות" משתפים‬
‫פעולה עם הסוכן‪ ,‬על‪-‬ידי כך שהם עוזרים ב‪"-‬מכירות" אחרי שמגיעים אליהם‪ .‬שיתוף‪-‬פעולה כזה יכול‬
‫לקרות בתרחישים מציאותיים רבים אחרים‪ ,‬שבהם ה‪"-‬לקוחות" שייכים למעשה לאותו ארגון כמו‬
‫הסוכן הנוסע‪ ,‬וניתן להורות להם לנוע כדי לסייע ב‪"-‬העברת הסחורה"‪ .‬מובן שעשויים להיות סוגים‬
‫נוספים של שיתוף‪-‬פעולה‪ .‬לדוגמא‪ ,‬יתכן שניתן להורות ל‪"-‬לקוחות" לנוע עוד לפני קבלת הסחורה‪,‬‬
‫כדי לפגוש את הסוכן‪ .‬בתרחישים מסויימים‪ ,‬שיתוף פעולה כזה עשוי להיות רלוונטי במקום‪/‬בנוסף ל‬
‫תנועה לאחר קבלת הסחורה‪ .‬כמו‪-‬כן‪ ,‬מטרות אחרות מלבד מינימיזציה של הזמן עשויות להיות‬
‫מעניינות‪ .‬לדוגמא‪ ,‬בתרחיש של הרובוטים שצריך להפעיל‪ ,‬אם לכל רובוט יש מצבר מוגבל אז עשוי‬
‫להיות חשוב לעשות מינימיזציה של המרחק המקסימלי שאיזשהו רובוט נוסע‪ .‬מטרה מעניינית נוספת‬
‫בתרחישים מסויימים עשויה להיות מינימיזציה של המרחק הכולל של כל המשתתפים )למשל אם‬
‫החברה שלהם משלמת עבור הוצאות הנסיעה הכוללות של כולם(‪ .‬מובן שכמו בבעיית הסוכן הנוסע‬
‫הקלאסית‪ ,‬עשוי גם להיות רלוונטי לעסוק בבעיה ללא הדרישה שהסוכן והלקוחות "יחזרו הביתה"‬
‫)יתכן שנקודת ההתחלה שלהם היא שרירותית‪ ,‬למשל כשמפזרים נחיל רובוטים בשטח מסויים(‪.‬‬
‫אפשר להתעניין בכל הבעיות הללו גם במרחב אוקלידי ולא בגרף‪.‬‬
‫כפי שמתואר בפרק ‪ ,3‬חקרנו את כל הצירופים של הגרסאות שהוזכרו למעלה‪ ,‬כלומר את כל‬
‫הצירופים של סוגי שיתוף‪-‬פעולה‪ ,‬פונקציות מטרה ומרחב מטרי‪ ,‬עם או בלי הדרישה ל‪"-‬חזרה‬
‫הביתה"‪ .‬הראינו שלרוב הבעיות הללו יש אלגוריתם שמשיג יחס קירוב קבוע‪ ,‬לרבות מהן יש ‪,PTAS‬‬
‫ולמיעוטן יש פתרון בזמן פולינומיאלי‪ .‬בצד הקשיות‪ ,‬הצגנו הוכחות ‪-NP‬קשיות והצגנו קבועים שלא‬
‫ניתן לקרב ביחס נמוך מהם‪ .‬חלק מתוצאות הקשיות שהצגנו הדוקות‪ ,‬כלומר תואמות את התוצאות‬
‫האלגוריתמיות שהצגנו‪ .‬כל האלגוריתמים שלנו הם קומבינטוריים לחלוטין‪ ,‬והוכחות הקשיות שלנו‬
‫משתמשות ברדוקציות מבעיות קשות ידועות‪ ,‬ללא צורך בשימוש במשפט ה‪ .PCP -‬הפרק הזה מבוסס‬
‫על המאמר ]‪.[AAS06‬‬
‫גרסת ה‪ min-max-‬של בעיית ה‪-r -‬אסיפות )‪(r-gathering‬‬
‫בפרק ‪ 4‬אנו עוסקים בבעיית ה‪-r-‬אסיפות‪ ,‬גרסה חדשה יחסית של בעיית מיקום‪-‬המתקנים ) ‪facility‬‬
‫‪ .(location‬בבעיית מיקום המתקנים הקלאסית‪ ,‬הקלט כולל קבוצת לקוחות ‪ C‬וקבוצת מיקומים‬
‫פוטנציאליים של מתקנים ‪ .F‬עבור כל מיקום פוטנציאלי נתון מחיר לפתיחת מתקן שם‪ .‬לכל לקוח‬
‫ומיקום פוטנציאלי של מתקן נתון מחיר‪-‬שירות‪ ,‬כלומר המחיר שיעלה לשרת את הלקוח ע"י מתקן‬
‫שימוקם שם‪ .‬הבעיה היא לבחור מיקומים לפתיחת מתקנים‪ ,‬כך שהסכום של מחירי הפתיחה ומחירי‬
‫השירות יהיה מינימלי )כל לקוח מקבל שירות מהמתקן הפתוח שמחיר קבלת השירות ממנו הוא‬
‫מינימלי עבור הלקוח הזה(‪ .‬הבעיה הזו מוצגת לעיתים קרובות כגרף דו‪-‬צדדי מלא‪ ,‬שצד אחד שלו‬
‫מייצג את קבוצת הלקוחות ‪ C‬וצידו השני מייצג את קבוצת המיקומים ‪) F‬כל אחד עם מחירו(‪ ,‬כך‬
‫שמשקלי הקשתות הם מחירי השירות‪.‬‬
‫‪ Hochbaum‬הציגה אלגוריתם קירוב לבעיה הזו שמשיג יחס )‪ .[Hoc82] O(log n‬לא ניתן להשיג‬
‫לבעיה הזו יחס קירוב )‪ ,[Arc] o(log n‬אלא אם כן ))‪ .NP ⊆ DTIME (n O(loglogn‬אם מחירי השירות‬
‫מקיימים את אי‪-‬שיויון המשולש‪ ,‬אז יש אלגוריתם שנותן קירוב ‪ 3/2‬לבעיה ]‪ .[Byr07‬מצד שני‪ ,‬לא‬
‫ניתן לתת לה קירוב טוב יותר מאשר ‪ ,1.463‬אלא אם כן ))‪ .[GK99] NP ⊆ DTIME (n O(loglogn‬רוב‬
‫המחקר עוסק בגרסה המטרית‪ ,‬שבה מחירי השירות אכן מקיימים את אי‪-‬שיויון המשולש‪ .‬בתרחיש‬
‫הזה משתמשים לפעמים במונח "מרחקים" עבור מחירי השירות‪ ,‬ואומרים שכל לקוח מחובר למתקן‬
‫הפתוח הקרוב אליו ביותר‪ .‬בעיית מיקום המתקנים המטרית נחקרה במשך עשורים רבים‪ ,‬תחילה‬
‫בעיקר בקהילת חקר‪-‬הביצועים‪ ,‬ויש לה גרסאות רבות שנחקרו באופן נרחב )ראו למשל סקירות של‬
‫]‪.([Dre95, Vyg05‬‬
‫כמה מהגרסאות החדשות יותר של בעיית מיקום המתקנים עוסקות באילוץ נוסף‪ ,‬על מספר הלקוחות‬
‫שכל מתקן משרת‪ .‬בבעיית מיקום מתקנים עם קיבולות )‪) (capacitated facility location‬ראו למשל‬
‫]‪ ,([Vyg05‬יש חסם עליון על מספר הלקוחות שמתקן יכול לשרת‪ .‬בגרסה עם "קיבולות רכות" ניתן‬
‫לפתוח יותר ממתקן אחד באותו מקום )כך שמחיר הפתיחה מוכפל במספר המתקנים שנפתחו במקום‬
‫הזה(‪ .‬בגרסה עם "קיבולות קשות""‪ ,‬בכל מקום פוטנציאלי אפשר לפתוח רק מתקן אחד‪.‬‬
‫עבור "קיבולות רכות"‪ ,‬יש אלגוריתם שמשיג יחס קירוב ‪ ,2‬בדיוק כמו ה‪ integrality-gap-‬של העידון‬
‫הלינארי של הבעיה ]‪ .[MYZ03‬עבור קיבולות קשות‪ ,‬יש אלגוריתם שמשיג יחס קירוב ‪5.83‬‬
‫]‪ .[ZCY04‬החסם התחתון הגבוה ביותר שידוע על יחס הקירוב הוא החסם התחתון של ‪ 1.463‬עבור‬
‫הגרסה שבה אין קיבולות ]‪.[SCY04, GK99‬‬
‫ראוי לציין שבשונה מבעיית מיקום המתקנים הקלאסית‪ ,‬לקוח לא בהכרח יקבל שירות מהמתקן‬
‫שמחיר השירות שלו הוא הנמוך ביותר עבורו‪ ,‬בשל אילוצי הקיבולות‪.‬‬
‫אחת הגרסאות היותר חדשות של בעיית מיקום המתקנים שמתחשבת במספר הלקוחות שכל מתקן‬
‫משרת היא בעיית ה‪-r -‬אסיפות ]‪ .[GMM00, KM00, Svi08‬הבעיה הזו היא אנלוגית לבעיית מיקום‬
‫המתקנים עם קיבולות‪ ,‬אולם יש בה חסם תחתון של ‪ r‬על מספר הלקוחות שכל מתקן משרת )במקום‬
‫חסם עליון(‪ .‬המוטיבציה הבסיסית היא שבדרך‪-‬כלל דרוש מספר מסויים של לקוחות כדי שפתיחת‬
‫מתקן תהיה כדאית כלכלית‪ .‬ניתן להסתכל על המקרה הפרטי של הבעיה הזו שבו אין מחירי‪-‬פתיחה‬
‫למתקנים גם כעל בעיית ‪ ,clustering‬שבה לא מעוניינים בקבוצות קטנות )ראו ]‪ .([AFK+06‬עבודות‬
‫קודמות עסקו גם במקרה שבו לכל מיקום יש חסם תחתון אחר על מספר הלקוחות הדרושים לפתיחת‬
‫מתקן בו ]‪.[GMM00, KM00‬‬
‫נשים לב שגם בבעיית ה‪-r -‬אסיפות לקוח לא בהכרח יקבל שירות מהמתקן שמחיר השירות שלו עבורו‬
‫הוא הנמוך ביותר‪ ,‬בשל האילוצים שיוצר החסם התחתון‪.‬‬
‫העבודות של ‪ [KM00] Karger and Minkoff‬ושל ‪ ,[GMM00] Guha et al.‬שהציגו את בעיית ה‪-r -‬‬
‫אסיפות‪ ,‬הציגו גם אלגוריתם קירוב דו‪-‬קריטריוני עבורה‪ .‬האלגוריתם שלהם מבטיח שכל מתקן‬
‫ישרת לפחות ‪ αr‬לקוחות‪ ,‬במחיר שגדול לכל היותר פי )‪ (1+α).β/(1-α‬מהמחיר האופטימלי לבעיית‬
‫ה‪-r -‬אסיפות‪ ,‬כאשר ‪ β‬הוא יחס הקירוב עבור בעיית מיקום המתקנים‪ .‬אם בוחרים ‪,α=(r- 1)/r+ε‬‬
‫האלגוריתם הזה נותן קירוב יחס קירוב של )‪) 1.5(2r-1+ε‬בקריטריון אחד כרגיל(‪.‬‬
‫לאחרונה ‪ [Svi08] Svitkina‬הציגה אלגוריתם שמשיג יחס קירוב קבוע עבור הבעיה הזו )יחס של‬
‫‪ .(558‬כמו‪-‬כן‪ ,‬עבור ‪ ,r=2‬העבודה של ]‪ [AK07‬הראתה שהבעיה פולינומיאלית אם אין מחירים‬
‫למתקנים‪.‬‬
‫העבודה של ‪ [AFK+06] Aggrawal et al.‬עסקה במקרה פרטי של גרסת ‪ min-max‬של הבעיה הזו‪,‬‬
‫שבה אין מחירי פתיחה למתקנים‪) C=F ,‬כלומר מיקומי הלקוחות זהים למיקומים הפוטנציאליים‬
‫לפתיחת מתקנים(‪ ,‬והמטרה היא להביא למינימום את מחיר השירות המקסימלי )ולא את סכום‬
‫המחירים(‪ .‬הם קראו למקרה הפרטי הזה‪ ,‬שהמוטיבציה אליו הגיע מישום ל‪ ,clustering-‬בשם‬
‫‪ .r-gather clustering‬בעבודתם הם הציגו אלגוריתם שמשיג יחס קירוב ‪ 2‬עבור המקרה הפרטי הזה‪,‬‬
‫והוכיחו שלא ניתן להשיג יחס קירוב טוב יותר עבור ‪) r>6‬בהנחה ש‪ NP -‬שונה מ‪ .(P -‬עבור הכללה‬
‫שבה מותר להתעלם מאחוז מסויים של הלקוחות שנתון בקלט )"טעויות דגימה" מבחינת ה‪-‬‬
‫‪ ,(clustering‬הם מצאו על אלגוריתם שמשיג יחס קירוב ‪.3‬‬
‫אנו מציגים אלגוריתם שמשיג יחס קירוב של ‪ 3‬עבור גרסת ה‪ min-max-‬הכללית של בעיית ה‪-r -‬‬
‫אסיפות‪ .‬אנו מוכיחים שלא ניתן לקרב אותה טוב יותר עבור ‪) r>2‬עבור ‪ r=2‬העבודה של ]‪[AK07‬‬
‫מוכיחה שהבעיה ניתנת לפתרון בזמן פולינומיאלי(‪ .‬בנוסף‪ ,‬אנו מוכיחים שלכל ‪ r>2‬לא ניתן לקרב את‬
‫בעיית ה‪ r-gather clustering -‬עם יחס קירוב קטן יותר מ‪ .2-‬בכך אנו משפרים את תוצאת הקשיות‬
‫של ]‪ .[AFK+06‬לאלגוריתם שלנו יש מספר הרחבות‪ ,‬ביניהן קירוב ‪ 3‬להכללה שבה ניתן להתעלם‬
‫מאחוז מסויים של הלקוחות )כלומר אנו משיגים אותו יחס קירוב לבעיה יותר כללית מאשר זו ש‪-‬‬
‫]‪ [AFK+06‬עסקו בה(‪ .‬בנוסף‪ ,‬אנו עוסקים בגרסה שבה על כל לקוח לקבל שירות מהמתקן הפתוח‬
‫שמחיר השירות שלו הוא הזול ביותר עבורו )"המתקן הקרוב ביותר"(‪ ,‬דרישה טבעית למדי‪ .‬אנו‬
‫מציגים אלגוריתם שמשיג יחס קירוב של ‪ 9‬עבור הגרסה הזאת‪.‬‬
‫לגרסה המקורית של הבעיה‪ ,‬שבה יש לעשות מינימיזציה לסכום המחירים‪ ,‬אנו משיגים יחס קירוב‬
‫של ‪ 2r‬במקרה שבו אין מחירי פתיחה למתקנים‪ .‬יחס קירוב כזה הושג במקביל לעבודתנו על‪-‬ידי ‪Lim‬‬
‫‪ ,[LWX06] et al.‬תוך שימוש באלגוריתם שונה‪.‬‬
‫התוצאות המתוארות בפרק זה מבוססות על המאמר ]‪.[Arm08‬‬

Exact and Approximate Algorithms for New Variants of Some Classic

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