Epsilon Nets and

15
EpsilonNetsand
Tlansversalsof Hypergraphs
More
For historicalreasons,
a finiteset-system
is oftencalleda hypergraph.
prccisely,
^ htperqraph
H consists
of a finitesety(il) of terticer(points)and
of E(fl) are usuallycalled
a family E(fl) of subsets
of y(H). The elements
(or, ft short,edgex).If the hyperedges
of I' arer-elementsets,then
hyperedges
l, is saidto be an /-rnrlormhlperyraph.Usingthisterminology,
a graphis a
two-uniform
hypergraph.
In Chapterl0 wehaveextended
somegraph-theoretic
(cf. Theorems
resultsto r-uniformhypergraphs
l0.ll and 10.12).
The conceptof hypergraphs
is a very generalone,so it is not surprising
in variouslieldsof
thathypergraph
theoryhasa largescaleof applications
l', a subset
I g y(fl) is
mathematics,
includinggeometry.
Civ€na hypergraph
for everyedgeE € E(H).Many
calleda tra s|e$al of H if ?n E is nonempty
andgeometry
canbe reformulated
as
extremalproblemsfrom combinatorics
queslions
in a
lransversal
of thefollowingtype:Whatis thesizeof a smallesl
givenhypergraph
H? Thisproblem,
in general,
is knownto becomputationally
(cf. Careyand Johnson,1979).However,undercertainspecific
intractable
conditionson ll, one can guarantee
the existenceof a relativelysmall
transversal.
The presentchapterfbcuseson resulhof this kind.In particular,
we shallseehow a powerfulprobabilistic
ideaof Vapnikandchervonenkis
geometric
and algorithmic
canbe appliedto obtaina numberof interesting
results,
TRANSVERSALSAND FRACTIONAL TRANSVERSALS
t€t fl bea hypergraph
withvercx sety(It) andedgeset,.(H).Letr(1{)denote
thesizeof a smallest
transversal
of fl, thatis, thesmallest
number7 suchthat
onecanchoose vertices
withthepropenythatanyedgeofli contains
at least
" is usuallycalledtherrdnrlendlnumber(ot thewrlet-cover
oneofthem.7(I1)
umber)of H.
Thepackingnumber(ot matchinq
number\of ahypergraph
l/ is definedas
thelargestnumber/ = /(li) suchthatfl has, pairwisedisjointhyperedges.
Obviously,
/(l1) < r(a) for anyhypergraph
H. Typically,
7(li) is strictlylarger
a
Epsilon Neti and TllNversrls of Hyperyraphs
+
rt
t
a
a
+
.i
a
a
than /(fl). In fact, 7(H) cannotevenbe boundedby any functionof I,(H) (see
Exercise15.3).
Let lll" denotethe setofall nonnegative
real numbers.l-et us call a function
l: y(A) -+ lR' a fractioaal transyersal of H if
(.r) > |
(15.t)
Sin
The minimumof >€v(fl) l(r) over all fractionaltransversals
of ft is calledthr
tactional transyersalnumberof H , andis denotedby r* (fl). Onecanassociate
with eachtransversal? of ll a functiont.: y(H) -+ ll{+ definedas
Ihu
)
",',-{l l[ili:
*
-
rD
.
t
.D
ID
lbr evervhvperedsef € E(H).
(15.1)and L. v(ol7(,r) = lll, we havethar
Sincethis functionsatisnes
t- (H\ <r(H).
Similarly,
afractionalpactrnSof Il is a nonnegative
functionp: E(H) -+ llt'
suchthat
p@\<l
for everyvertexr c Y(i1).
ihe
-'lu
:e\
<D
t
(-
tC
.-
The maximumof >.e.(fi )p(E) over all fractionalpackingsof 11 is calledthe
fuactiokolpackingnunber of 11,and is denotedby /"(H). As before,we hale
v'(H) > v(H).
It is easy to deducedirectly from the definitionthat /'(fl) < i'(H) (see
Exercisel5.l). In fact, thesetwo numbersare always equal to each other.
Moreover,the following is true.
it
Th€orem 15.1. For e,reryhtpergraphH,
t
rt
-
-a
--
v(H) < Y'(H, = r' (H) < 'r(H),
l
:,n
l,'f
Th€
P
and the valueof r'(H\ = r'(H) can be determinedby linear pmgrammiftg
-a
-+
-t
-'a
+
Proof, Let,ri (1 < i < r,) and tJ (1 <j < ,.) be the verticesand rhe edges
of 11,respectively.
Let A = (aij\ be the incidencemattix of H, i.e.,
= | 1 if rieEj,
",i lo r ,,erj.
Let A7 denotethe transpose
of A, and let 1,,denotethe matrixconsisiingof one
columnof lengthr, all of whoseentriesare 1's.civen a functionr: y(H) ) li
(^nd p: E(H, + lli), let I (rcsp.p) denotea matrix consistingof one column
whoseith entry is r(ri) (resp.p(E-)).
..n
and T.amvebrlsof flJp€rgnphs
by anyfunctionof /(fl) (see
I}msv€ruols and lrrctiond T.lDsveBlls
Observethat t is a fractionaltransversalof 1{ if and only if
rmbers.
Let uscalla function
CeE e E(Hr.
( l s .)r
ranxversals
ofH is calledthe
lby r'(lt). Ooecanassociate
-+ Il* definedas
,ttlx) = lTl, w€ have rhal
ativefunctionprE(/t) + ill'
( ir € Y(It).
I packingsof fl is calledthc
r, ,'(l/). As before,we have
)n that r'(ft < r'(I/) (see
rlways equal to each other
245
A7l>1,, and !>0.
Similarly,p is a fracrionalpackingof 11if andonly if
Ap<1,, ^nd p>O,
Thus,
r'(I1) = min{llllArl > 1,,,! > 0},
v'@)=n x {rIp_lAp< thp>O}.
problems
Thesetwo linearprogramming
aredualto eachother,so it follows
immediately from the duality theorem of linear prograrnmingthat their
solutions,
r'(f4 and r'(tl), areequal(see,e.9.,Papadimitriou
and Steiglitz,
1982;Chvital,1983;andCrijtschelet al., 1987).
n
In general,
r'(l/) canbe muchsmallerthan7(fl) (seeExercise15.3).The
followingtheorem
of Lov6sz(1975)showslhatlhisis ooithecasewhenevery
pointof lJ belongsto relativelyfew hyperedg€s,
Theorem 15,2 (InvAsz), kt H be a hypergraphwhoseerery ve ex is
conlainedin at fiost D edges.Then
-l
7'(lt) < r(H) < (ln D+ I)r'(fl).
H),
by linear Wogmmming.
, the vertices and the edges
Proof. We haveto proveonly the secondinequality.Let t: y(fl) + R+ be
a fractional
transversal
of fl with L€vrr) (.r) = r'(fi).
We arc goingto selecta setof venices.rt,xz,...by a greedyalgoithm.
Let xr b€any ve(exof H whosedegree(i.e.,thenumberof edgescontaining
it) is maximal,Let Dt denotethe degreeofxr in lt. Setllr = FI .rt, thal
is, the hypergraph
obtainedfrom H by deletingth€ vertex.rt andall €dges
containing.rt.If-r1,...,ii
e y(A) havealr€ady
beenselected,
thenlet Ilr =
H xt - t2 - ... - ri. lf Hi hasno edges,we stop.Otherwis€,
let ri*r be a
vertexof l/i whosedegreeDr*t is maximal,andso on.Clearly,
h€ matrix consisting of one
en a function.: Y(It) ) lll
consistingof one column
lE(H)l - lE(Hi. t)l= Di.t.
By theproperties
of ,,
(15.2\
aa
.l
-a
2$
F4sllon Neb.nd ftrl|lveNk
ol Hyp.rgnpls
lE(ri)l=
> r<Ee>gHl
Ee ElHt)
=I<'rI
r<YlHl
< )
I
Ee E(Hl
Etr
(')D,-'
3 Di+tt'(Hr.
Assumenowthatour proccdureterminaiesin s steps;i.e.,t , is empty.Then.
of course,
r(H) < r. PutII0 = H. By (15.2),we h.ve
,= -l,/ L / n = S lE(t/)l-lE(I1ar)l
r
tE#r)l
_ ___F_
=
+,\-,.,-,,r
).p\nt1ll'\
'!u'
ot
D*r
_r\
Dt l'
Hence,usingtheinequality
lE(A)l < Di+ri"(fl) (0 < j < s) andthefactD" > l.
we obtain
" <"1a1*!a,-,"'{r,(E;i;)
= , ' '1\ a
s y+*1D!1, J D , a * ' 1
.r,u,('-;-=t.,
+)
=",r"r(r._j,f;
< r'(H)(l + ln Dr).
Thus,
7(fl) <s <7'(g)(l + ln Dr),
as desired.
siton Netsmd lianNe6ds of H}lergraphj
<s
r(j)
.,s
I
r )Di+|
L
inatesin r stepsii.e.,iI* is empty.Then,
i.2), we have
,, )l lE(H,+r)l
D*t
,
I
I\
'' \ o*,='
a)
.r7 (H) (0 < i < s) andthefactDr > I,
.
;rnt\
I
^
,
,, D,+r\
DI
tlr
t\
r)
t\
g)
Vapnil-Che.voneDl*
u7
DimeBlon
VAPNIK-CIIERVONENKTS DIMENSION
Suppose
that for a publicopinionpolt we wantto selecla smallnumberof
individualsrepresenting
all majorsectionsof the society.First,we haveto
choosecerlain categoriesof fEople and then decidewhich of thesegroups
arc considered"importaot."Accordingto our democraticprinciples,we sh;lt
measure
the "importance"
of a groupby its size(in the percentage
of the
population).
Thenthe importantgroupswill definea hypergraph
fl with the
propenythatlEl > €ly(fl)l for everyedg€E € E(li), where€ is someflxed
(0 < € < l). Thesmallest
conslant
numberof peoplerepresenting
all important
groupsis r(fl).
Clearly,th€ functiont(xt = l/(elv(H)l'r,for att, € y(H), is a fractional
traNversalof F/ with >.vott(x) = l/e. Hence,r'(fl) < l/e, andTheorcm
15,2imDliesthat
r1a;<11tna+11,
(rs.3)
where, is themaximumdegree
of thevertices
of It. Thisboundis extremely
poorif D is larye.
In theirseminalpaper,VapnikandChervonenkis
(1971)pointedout thatif
It satisnescertainoaturalconditions,the aboveupperboundcan be replaced
by a functiondepending
onlyon €. To specifytheseconditions,
we need_
some
pfepamtron.
Defnitiarn 15.3. Let H = (V(H),E(H, denote a hyp€rgraph.A subset
4 E y(tf) is caffedshattererl
if for everyB C ,4 thereexistsan E € E(Il)
suchthat En A = 8. Th. Vapnik-Chenonenkis
dintension(or VC dime sion\
of H is thecadinalhyof rhelargesr
shattered
subser
of V(H).Il will bedenorcd
by vc-dim(Ii).
'Thefollowing
theorcmwasprovedindependently
by Shelah(19?2),Sauer
(1972),andVapnikandCh€rvonenkis
0971).
Theorem15.4. Let H be a hyperqtupllwith n yertice!and Vc-dimensiond.
The
l r t s r l s/, ,1r\,+i,/,1, \+ * ( a l
and this bound ca not be improved.
)(l+lnDr),
First Proof, The assertionis tdvial ifd = 0 or', < d. Assumethatwe have
akeadyprovedil for every hypergraphF wilh Vc-dim (H-r< d. and lbr every
-la anO
hypergraph
H wirhvC-a;mrlir
lytntl< z.
Givena hypergraph
H with', vertices
andVc-dimension
d, letusdefinetwo
--
I;psilon Nc(g and Tta.sv€rsals ol Hypelgraphi
e-J
otherhyperSraphs,
llr and H2, as fbllows. Let V(H t) = V(H) = V(H)
for somenxed .r e Y(Ii), and set
a-
Irl
E(Ht)= lE UllE e E(H\\,
E ( H ) = l E e E ( H ) l rd E a n d E U h l e E ( H ) ) .
-l-
--
rD_
.D
-)
.-
Obviously,
Vc-dim(Hr)<dandVc-dim(dz)<l
l.
On theolherhand.by rheInduct,on
hyporhesis.
lE(H)l= |E(Ht)l+ lE(Hr)
st/".'\*!/".'\
!,r\tt4\tl
<
a-
=t/'r\
-
.fa
a-
i)
.l-
-a
t-
tID
'Ir
-t
..-
The tighlness of tni* mun,t follows from the lrcr that il ,(H)
(rr)=/.
vc-dim
{Uc YI lUl< /}, then
We also includea slightly more complicaledproof due to Frankl and P!!h
(1983),becNsc it is a good illustrationof the so-c led line$ altehru netht\;
(see,e.9.,Bab.i AndFrankl, 1988).
SecondProof. Let E(fl) = {E,l I < i <,r}, and let Xr. | < j < >11,,(1 t. b"
a list of all subsetsof y(lt) of size ar most d. Definean ,, x >'/.{, ('jJ marn\
6 = (a,) b]
Il
,r=10
iff,rx.
if E, b x,.
Suppose,for contradiction,that ,i > Z;=.1;l lhen lhe rowror A .,r.'
Iinearly dependentover the reals; thus there existxa nonzerofunclio:
/: E(fl) ) .ll suchthat
) rtar=o
for everyXr.
LEt A e V(H) be a ,ninirnclsubsetfor which
-
l./ta,t="+0.
(SetsA with nonzerosumscenainlyexist, for we get a nonzerosum fbr xnl
ma\imal elementA of the family {A e 6(A)l/(4) I 0}.) Obviously,lAl > /+ l
Given any B g A, let
r and Transveredsof Hyperyraphs
vlpnik-CheNonenkis Dimosion
v(H)=v(H)=V(H)-I')
F(B)= tL / " /(8,).
EiaA-B
Thus, F(A) = d, and setting I = A - {r} lor any fi xed d e A,
EU tile E(I1)].
id
is,
r,p\- S
L
rir.,
Ei2B
S
r,r\
EiaA
=0-a=-a
In general,
if I is any(lAl- t)-elementsubs€t
of A (0 <,t < lA ), rhen
/,-l\
F(B)=(-t)td+0.
This yields,in particular,thatthereexistsat leastonehypercdge
Ei with EinA =
B. Thus,A is shatter€d,
contradicting
our assumption
lhatVc-dim(fl) = d. O
n the fact that tf E(H)
=
n
proof due to Frankland Pach
)-calledlinear algebramethod
nd letxj, 1 < j < >':{,(';), be
)efineanmx >Lo{'l) matrix
(1971)discovered
Vapnik and Chervonenkis
an ingeniousprobabilistic
(counting)
argument
basedon the abover€sult,whichleadsto a substantial
improvement
of the bound(15.3),They showed(in a somewhat
different
setting)that thereexistsa function/(d,€) suchthat the transversal
number
of everyhypergmph
11of Vc-dimensiond, all of whoseedgeshaveat least
€ly(I/)l €lements,
is at most/(d, e) (seeExercise15.6),Theideasof Vapnik
andChervonenkis
havebeenadapted
by Haussler
andWelzl(1987)andBlumer
et al, (1989)to obtainvariousupperboundsod /(d,€). Theseresultswere
sharpened
(1992),asfollows.
andgen€ralized
by Koml6s,Pach,andWoeginger
Civen a finite s€t y, a functionp: V + ll{+ is c lled a probabilitymeasureif
= l.
tL . - - at.r)
'
:t,
l). Th€nthe rows of A are
e existsa nonzerofunction
Themeasure
of anysubset
X g y is definedby p(X) = >€ x t(i)
Theorem 15.5 (Komf6set al.). Iet H be a hypergraphof Vc-dime siond,
let . > 0, and let p be a probabilil! measureon V(H) suchthat p(E) > Efor
everyE e E(H).Thenr(H) < t(d,.r, wherct(d,e)denotes
positive
thesmallest
integert satislti g
z!/lrir
_.J\it\.
.1)''-"'.r
rt
tly,for atryE< i, wehave
for someintegerT > t. Conseque
we get a nonzero sum for any
l) / 0).) Obviously,
lAl > d+ 1.
,vttt!(nl+zrnrn 1* o\
€\ €
e
(cf. Exercise15.9).
/
250
EpslloDNeh rnd llansveru€ls ot Hypergrrpht
Proof. Let us selectwith possibl€repetitionr mndompointsof y(H),
wherethe selectionsare donewith resp€ctto the probabilitymeasurer. We
geta rundomsample
t e l v ( H ) 1=|v ( H , x
xv(H).
we saythat r is a traflrveEol ot H if ewry edgeE € E(l/) containsat least
onepointof r, Let /(8,.r) denotethenumberof components
of r thatbelonS
Thcn
to E, countingwith multiplicity.
Prtr is not a [ansversal
of fl] = Pr[3E € E(Hrt I(E,x)=01.
Havingpickedthe stringr of lengthl, let us chooserandomlyanotherT I
efements
from V(H\. Lgt , I IV(H\lr1 denotethis new stdng,and let .
Furth€rmore,
let (z) = (ry) denotethc
.r) e [V(,/)lr standfor the full sequence.
occurring
in z (i.e.,theyarccounted
with multiplicilicr
multiretofall elements
but theirorderis irrelevant).
ForanyE € E(It), l(E,t) is a randomvariable
havingbinomialdistdbul;on.
Let mEbe the medianof I(E,t ,
PrII(E,9 > '/l.|l< + <tulI(E,y, > nd.
The following inequalityis an immediatecons€quence
of the independencc
of .t and,.
Pr[3 E € E(r4 r1(E,i) = 0]
Prlf E € E(It) : /(8,.t) = 0 and/(E,)) > t'tEl
h [/(8,]) > l,rEl
Emind)
< 2PII]E e E(Hr:I(E,,
*t
0 andI(E, )) > |'lE1.
For a fixedt € E(H), the conditionalprobabilityfor given(z) = (xy)
EpsiloDNels rnd TraEeeNh of ttyF.s.aph,
v.pn|t{tenondlir
DiDeBioD
)ssiblerepetition/ randompointsof y(I1),
lh rcspectto the probabilitymeasurcp. We
Prt 1(Er) = 0 and/(t' )) > 'rEl(z)l
(!:'\
= xlt\E,z)
> nElYgL
"l r
f/ if every edgeE e t(I/) conlainsat least
lhe numberof cohponentsofr that belong
'f r/l= Prlf E€ E(H):r(E,x)=Ol.
\
\r(E,z)
I
= V(H)x,,.x V(H).
--_/-
>.ut(r - +)""'
<xt1(8,.)
<xu<r,zt>nd(t
l)'''.
(Herex [A] is thecharacteristic
functionofA, thatis,x [A] = I ifA is true,and
0 otherwise.)
J-,r
By TheoremI5.4.a 6xedmultiset(z) has at most-\).{ I.I different
I
wirhrh€€dgeso[ 'y'.Thus,
intersections
PrllE e E(H\tI(E,r)= 0 and1(8,))> n,I (.)l
:rh /, let us chooserandomlyanotherI _ t
atll '_denotethis new string,and let z =
uence.r.urlhermore,
Iet (z) = (.x))denot€the
n : (r.e.,mey arc count€dwith muhiplicities
ndomvariablehavingbinomialdistribution.
'ritf' +r-'
4-l
where,n = minE€r(r) mr, Usingtheknownfactthatthemedianof a binomial
distribution
is within I of themean,
I > (7- 0€- L
n > (T- t)smind)p(E)-
< I < Pr[/(8,])> rrrl.
rmediateconsequence
of the independence
0j
Hence,we obtain
tl
't' t.
p r t f E € E ( H ) : r ( E=. r0) 1< z 1f f r ) t / r ' 1 1 ' 7
r J /\ '
1/
':l
of ly',with positive
is lessthanl, thent is a transversal
If the lastexpression
probability.
of thetheorem.
Choosing
Thisprovesthefirststatement
1 =Ll€4\f 6 1e * 2 t n 61e 1 6/)- )l ,
€;l
"' - l L . i . l '
/r E,r) = 0 and,ttE.y) > rr! I
FiiEy)),.t-
we get after some calculations that
I pn,babilityfor given (2) = (ry)
-1;'''.'.',
'i11;1'
providedthate< j.
tr
Epsilon N€ts and Td6v.rsrts
of tttpcrgrophs
The abov€theoremis valid for any probabilitymeasure
p dcfinedon rhe
vertexsetof A. In particular.
onecanchoose
p to be constanr;
thatir, p(_r)=
l/lv(H)l fot evety:r€ y(tt). Wecandeduceanorherinrere$ring
Lesultfrom
Theor€m15.5by applyingir to the measurep,(x) = tlxr/;,@), wherc
t y(11) ) llt" is I fiactionalkansversal
ofH wittr l,.",,r,r1xj _ r-1f;.
Observe
thatin thiscase
?_-
p\rt=
sr
,
Ztt
t^l
(-t)
I
=s
->
A' r'(H) r"(H'
h:'l,i)r:r e"v:iy[ € E{Hr.Thur.choo}ing
c
oDril,nthc l(Jllowtng.
a
a
-
a
a
tl
a
a
a
a
a
Q
t/, {H) i Thrt,remt5.5.\rc
Corollafy 15.6(Komf6ser ^1.). kt H be a y hlperynph ofVc-dn,k,tsiot d.
(i, If ewtl ctlsc of H hat at tcast.lv(Htl etentcnts
ft)t rionc E s \, thtn
^h < !(h
€\
1*zrnrnl*g.
ii, Ar'@)>2. n ,l
r(H) < dr'lH)(tn r'(H) + 2 tn ln 7'(H) + 6).
Next we show rhat lbr./ > 2, the boundgiven in Theorem 15.5is chse to
.
Derngopttmal.
Theoreml5.T r Kumlcj\et li.t. Cive nry Mtnr. t ttt,erJZ2n ,ttryrL,t!
1< 2/U + 2L rh.rc ct(i\ts.t ' ottsh u t.t t > 0 u irh rhel,,lk\ ins U4,L.r!\,
rrl n,,\ € s c,l r. nr,.,( ttl coutrur! d hrp, rqmt,tt H il Vc-lruk u:kat ,t, tl
t'l [th,.\c.Juc\ lh^e nt k.N rlvtHtl p,,i ts.nn,t
.tH)>(.t- 2+"i+h _l
€
Pmof. A8rin, we usethe probabilislicrnerhod.L€l.y, be a lixed consranl.
1 , . <1 < 2 / t J + 2 t . C i v e na s u f f i c i e n lsl ym a l cl . t e t , l t K l a j l n { t / r r . w h e r . .
x r\:r conshntdependinS
only on d. l. and ), {burnot on aJ,whichwrll h.
specifiedlarer.Funhermore.ler
t=(d
(;)'
2+.iLh
1
€
We assumethat ,1.r, I are integers,disregarding
all roundol-feooa.
f,psllonNe6 .nd lrlnsreBds ot Hypergraptb
' fiy probability
measurep defined on the
can chooAep to be constan(that is,
r.(J) :
can deduceanotberinterestingresuh frorn
tie measurep'(tt = t(x)/r.(H), wherc
rsversalof H with
>.€ vU|r{x, = ,+{l/).
V.Dnik-Chenondlh
Dine$loD
t-et y be a fixed n-elementset.Constructa hypergraphI{ on the vertexset
y by randomlysel€cting
somer-element
subsets
of y, whereeachr-tupleis
with probabilityp. we arc going to showthat with high
chosenindependently
probability
(t) vc-din (It) s d, and
, u'lx)
(ii) '(it) > t.
.r(r)>l
'l t'tHt - f (H)
> dl
PrIVC-dim(H)
hoosing€ = lh'(fl) in Theorem15.5,we
et H beanylrypergmphof Vc-dimension
d,
I EIV(H)Ielements
fo\one e< !, then
-I+l 2 l n l n
'(H)
-+6).
subset,4
by l/l
< (, 1, ) prla nreata + l)-element
c y is shattered
\d+l/
e (H)tE.rA=B]
= f , 1 , ) f J p r t 3 EE
AcA
=(,1, ) II fr- (r-Pfrrlsl))
J+|
, ' ..,-- , '' . '. )
=(d- , ) I I (' - (' - p)'
' )
'j=o
+ 2hln r'(H) + 6).
boundgiven in Theorem I5.5 is closeto
=(,1') fif ' (r-p)(.'"''l'))('11")
?nany natural number d 2 2 and any real
.,1}.>0 tvith thefolbwing prcpen .
=f ''') rI(t-(r
Py'"];1'l'))r
rr\
\(+r/
,
, 'f ,r.dimension
d.a,
;.il::,:ffu
:*1;11n l.
'i')
-. t\ d +, l / \ I ri t+, \r'l\rr--dd
€€
listicmethod.
Let7, bea fixedconstant.
l\ small€. ter, = ((/€)tn(t/€r,where
t. and?' fbut noron €), whichwill he
t
\
t n - d - l \ t , n - d - l ,r' l
[a* rjPf'
d-t)\P\
,
d )
r
r
-<' , J +r' ,\ /r,)I\ n1l 1 t ' t 1 , 1 n 11 l J i r - '
\'\'/\lr/
/
?
t = ( d - 2 + . y' €t ! h ! .
n
r \{il')
l+i/
l\J+l^
= ( r (I n : J
., ^ ,
e! (t+.n,
a
,regardlngall roLudofferrors.
whichtendsto 0 ast ) 0. This prov€s(i).
Next we showthat (ii) also holdswith high probability.
ID
DpsilonN.ts md 'Irunsvcbals ot llyp.rsr.phs
254
1-
'I
P,h(It)< 'l = (n (r - pl";')
)
(.-
'(l)*'[-(",.')]
rl
.(#)'*o[,(:)(' ;j- )']
II
-
rO
,t
F r o mt h i s ,u s i n gt h ei n e q u a l i tI y . r l . r > " t " f o r b > 6 , 0 < r < l / a .
we obtainthe upperbound
^"''-'']
*o [-r(',1)
(+)=
"
tL
-
jf
1-
1-
- tl- 1'+K(d 2+.i/lK
1-
O
-
t
|l
-
(t
ta
-
O
(D
) ' . , , n r e ' |r ) " ^ i ' : ' , r / r ^
which tendsto 0 if
ID
{t
l/h,
d)<-1,
The conditionr/ > 2 in Theorem15.7is not merelya technicalassumption.
In frct, it is not hardlo oharacterize
allfnite hypergraphs
H with VC-dimension
I, andonecancheckthat.r(H)<fl/e1 I, providedrharcveryedgeofH htls
nt lerst eLv(/t)l pointsfbr some0 < € < I (seeExercise15.8).
The lollowing simple asse(ion will help us in decidingwhethera given
hypergraphhos low VC-dimension.
L€mma 15.8, Llt H be a hfpcrefttph oJ Vcaln ensio d, d d k!
p(Et,...,Er) be a sct-thcorctic
funnla dk ratithles(uxinlU.n, ). tfcre^.
ulge E of a hjpergnrylt H' can Irc ettprcrsedas
E' = p(Et, . . ., Er')
for suitabb Et e E(H),
then
Vc-dim(fl') < 2./klog (2./l).
Prool Let A be a d'-elementsubsetof V(H') = V(H), which is shatrered
in A'. By Theore'n15.4,
r{EnArEer,",rr.i(1')
Usingthea\.umpronon H'. lhi\ yields
Nre.sds of Ilyp$8rryhs
Rrnse Spacesand €.Nets
=t{E:
nAtE ea",rrr.1i(11)2'1'
'
w€ obiBjnthatd < 2dkloEQdk),
thetwo sidesof thisinequality,
Comparing
tr
as requircd.
\,'l
t+l ) J
> a,o <x < l/o- l/b,
anolherpaometerof
Ding,Seymour,
andwinkler (1994)haveintroduced
closelyrclatedto its Vc-dimension.
fiey defin€dI(H) as the
a hypergraph,
largestintegerI suchlhatonecanchooset edgesEt,82,..., Er € E(It) with
th€ prop€rtythat for any I < i <j < I, lhere is a vertexit e E; O E; that
doesnot belong!o any otherE* (s I ij), tt is ersy to se€that Vc-dim(fl) <
s
H. Combining
Corollary15.6withRamsey
for everynypergraph
( ""',
)
thefollowingresult.
lheor€m(Th€orem
9.13),onecanestablish
Thmrem 15,9(Ding et al.,1994), Fot anf hyperyraphH,
<
I,
tr
/ a technicalassumption.
rhsH with vc-dimension
that every edge of I/ has
ise 15.8).
eciding whethera given
'-dimension
d, and lel
: (u$ingU,i, -). If every
\(d.).11(1t)
\'.
'(n) < 6r1n)(r(rr)+ v(r))/
\ x(fi) )'
it is impossible
ofthischapter
wepointedoutthalin g€neral
At thebeginning
to boundr fromaboveby anyfunctionof ',. Gyidis andLehel(1983,1985)
ioitiated the investigationof certainclassesof hypergraphsfor which such
functionsexist.Theorem15.9providesa sufficientconditionfor a family of
K
Il impliesthatif thereexistsa constant
hyp€rgraphs
to havethis property.
suchthat \(E) < ,< for all memb€rsof a family, thent can be bolrndedfrom
of thisfact,
of /. For variousgeometric
cons€quences
aboveby a polynomial
seePach( 1995).
RANGESPACESAND e"NETS
, e E(Hr,
Y(H), which is shattered
)
of the
the relevance
Hausslerand Welzl(1987)wercthe 6rct to recognize
abovemachineryto g€ometricproblems,andin fact theyfomulatedandproved
of the
the first versionof Theoreml5.5, too.It seemsto capturetheessence
so-c lled random(or probabilistic) methodin a large variety of geometnc
kit will saveus a lot of time (and space)
applications.
This ready-to-use
in situationswherc otherwisewe would go through lengthy but foutine
calculations.
However,the mainsignificance
of theseideasis thattheyshed
somelight on the generaltransvenalprobl€m.The transversalnumberis a
glabal para''x.etet
of a set system.The resultsin the precedingsectionshow
that in any mcasurespaceof total measurel, any systemof largemeasurable
providedthatils rocdlbehavioris nice
setsadmitsa relativclysmalltrunsversal,
(i.e.,its Vc-dimension
is bounded).