Identifying codes in hereditary classes of graphs and VC-dimension

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VC-dimension

Nicolas Bousquet, Aurélie Lagoutte, Zhentao Li, Aline Parreau, Stéphan Thomassé

To cite this version:

Nicolas Bousquet, Aurélie Lagoutte, Zhentao Li, Aline Parreau, Stéphan Thomassé. Identifying codes

in hereditary classes of graphs and VC-dimension. SIAM Journal on Discrete Mathematics, Soci-

ety for Industrial and Applied Mathematics, 2015, 29 (4), pp.2047-2064. �10.1137/14097879X�. �hal-

01038012v2�

NICOLAS BOUSQUET

† k

, AURÉLIE LAGOUTTE

‡ ∗∗

, ZHENTAO LI

§ ††

, ALINE PARREAU

¶‡‡

,AND

STÉPHAN THOMASSÉ

‡ ∗∗

Abstrat.

Anidentifying odeof a graph is a subset of its verties suh that every vertex of the graph isuniquely

identiedbythesetofitsneighbourswithintheode. Weshowadihotomyforthesizeofthesmallestidentifying

odeinlassesofgraphslosedunderinduedsubgraphs. OurdihotomyisderivedfromtheVC-dimensionofthe

onsideredlassC^{,thatisthemaximum}VC-dimensionoverthehypergraphsformedbythelosedneighbourhoods ofelementsof C^{. Weshowthat hereditarylasseswithinnite}VC-dimension haveinnitelymanygraphswith an identifyingodeof sizelogarithmi inthenumberof verties while lasses withnite VC-dimension havea

polynomiallowerbound.

Wethenturntoapproximationalgorithms. WeshowthatMin IdCode (theproblemof ndingasmallest

identifying ode ina givengraph from somelass C^{) is}log-APX-hard for any hereditary lassof innite VC- dimension. For hereditary lasses of nite VC-dimension, the only known previous results show that wean

approximate Min Id Codewithin a onstantfator insomepartiularlasses, e.g. linegraphs, planar graphs

andunitintervalgraphs. WeprovethatMinId Codeanbeapproximatewithinafator6^{forintervalgraphs.}

Inontrast,weshowthat MinId Code onC4^{-free bipartitegraphs(alassofnite} VC-dimension)annotbe approximatedtowithinafatorofclog(|V|)^forsomec >0^.

Keywords. Identifyingode,VC-dimension,Hereditarylassofgraphs,Approximation,Intervalgraph

AMSsubjet lassiations. 05C69,05C85,05C62

1. Introdution. Let G = (V, E) ^{be a graph. An}identifying ode of G ^{is a subset} C ^of

vertiesofG^{suhthat,foreahvertex}v∈V^{,theset ofvertiesin}C ^{atdistaneatmost1from} v^{, is non-empty and uniquely identies} v^{. In other words, for eah vertex} v ∈ V(G)^{, we have} N[v]∩C6=∅⁽C^{isadominatingset)andforeahpair}u, v∈V(G)^,wehaveN[u]∩C6=N[v]∩C

(C ^{is a separating set), where} N[v] ^{denotes the losed} neighbourhood of v ⁱⁿ G ⁽v ^{and all its}

neighbours). WesaythatasetX ^ofvertiesdistinguishes u∈V(G)^fromv∈V(G)^ifN[u]∩X 6=

N[v]∩X^{. Thisoneptwasintroduedin1998byKarpovsky,}ChakrabartyandLevitin[21 ℄and hasappliationsin variousareas suh as fault-diagnosis[21 ℄, routingin networks[23 ℄ oranalysis

ofRNAstrutures[19 ℄. Foraompletesurveyontheseresults,thereaderisreferredtotheonline

bibliographyofLobstein[24℄.

Two vertiesu^and v ^{are twinsif}N[u] =N[v]^{. Thewhole vertexset} V(G)^{is an}identifying odeifandonlyifG^{istwin-free. Sinesupersetsof}identifyingodesareidentifying,anidentifying odeexistsforG^{ifandonlyifitistwin-free. Anaturalprobleminthestudyof}identifyingodes is to ndone of a minimum size. Given a twin-free graph G^{, thesmallest size of an}identifying odeofG^{is alled the}identifyingodenumber of G^{andis denotedby} γ^ID(G)^{. Theproblemof}

determiningγ^{IDisalledtheMin Id Codeproblem,anditsdeisionversionisNP-omplete [8 ℄.}

LetX ⊆V^{. Wedenoteby}G[X]^{thegraphinduedbythesubsetofverties}X^{. Inthispaper,}

we fous onhereditary lasses ofgraphs, that is lasseslosed under taking indued subgraphs.

Weonsiderthetwofollowingproblems: ndinggoodlowerboundsandapproximationalgorithms

fortheidentifyingodenumber.

1.1. Previouswork. Inthelassofallgraphs,thebestlowerboundisγ^ID(G)≥log(|V(G)|+

1)^{, sineall thevertiesof the graphs havedistint non-empty} neighbourhood within the ode.

Monel[27℄haraterizedallgraphsreahingthislower bound. Asforapproximationalgorithms,

†

LIRMM,UniversitéMontpellier2,Frane

‡

LIP,UMR5668ENSLyon-CNRS-UCBL-INRIA,UniversitédeLyon,Frane

§

ÉoleNormaleSupérieure,Paris,Frane

¶

LIRIS,UMR5205,UniversitéLyon1-CNRS,Frane

k

DepartmentofMathematisandStatistis,MGillUniversityandGERAD,Montréal,Canada

∗∗

PartiallysupportedbyANRProjetStintunderContratANR-13-BS02-0007.

††

PartiallysupportedbyaFQRNTB3postdotoralfellowshipprogram.

‡‡

PartiallysupportedbyaFNRSpost-dotoralgrantattheUniversityofLiège.

thegeneralproblemMin Id Codeisknowntobelog^{-APX-hard [22 ,23 ,33℄. Inpartiular,there}

is no (1−ε) log(|V|)-approximationalgorithm for Min Id Code. The problem Min Id Code remainslog^{-APX-hard even insplit graphs, bipartite graphsor} o-bipartitegraphs(omplement ofbipartitegraphs) [14 ℄.

Onthepositiveside,therealwaysexistsaO(log|V(G)|)approximationforMinIdCode[33 ℄.

Moreover, evenif in thegeneralaseMin Id Codeis hardto evaluate, thereexist several on-

stantapproximationalgorithmsforrestritedlassesofgraphs, suh asplanargraphs [29℄orline

graphs[15 ℄.

For the remainder of this artile, n ^{denotes the number of vertiesof} G^{. Table 1 gives an}

overviewoftheurrentlyknownresultsforsomerestritedhereditarylassesofgraphs. Theorder

ofmagnitudeofalllowerboundsarebestpossible(thereareinnitefamiliesofgraphsreahingthe

lowerbounds). Min Id Codeforlinegraphsandplanargraphshavea polynomialtimeonstant

fatorapproximation algorithmwith thebest known onstantwritten in parenthesis. Fromthis

table,weobservetwobehaviours: alasseither

1. has a logarithmi lower bound on the size of identifying odes, and Min Id Code is

log-APX-hardinthislass(forexamplesplit, bipartite,o-bipartitegraphs),or

2. there isa polynomial lower-bound onγ^ID(G)^{and aonstantfator}approximationalgo- rithmtoomputeγ^ID(G)^.

Graphlass Lowerbound Complexity Approximability Referenes

Allgraphs Θ(log(n)) ^NP- log-APX-hard [21 ,22 ℄

Chordal Θ(log(n)) ^NP- log-APX-hard [14 ℄

Splitgraphs Θ(log(n)) ^NP- log-APX-hard [14 ℄

Bipartite Θ(log(n)) ^NP- log-APX-hard [14 ℄

Co-bipartite Θ(log(n)) ^NP- log-APX-hard [14 ℄

Claw-free Θ(log(n)) ^NP- log-APX-hard [14 ℄

Interval Θ(n^1/2) ^{NP- open [16 ,17 ℄}

Unitinterval Θ(n) ^{open PTAS [13 ,16 ℄}

Permutation Θ(n^1/2) ^{NP- open [16 ,17 ℄}

Linegraphs Θ(n^1/2) ^{NP- APX(4) [15 ℄}

Planar Θ(n) ^{NP- APX(7) [3 ,29 ℄}

Table1

Knownlowerboundsonγ^ID(G)^andapproximabilityofγ^ID(G)^.

1.2. Our results. The aim of this paper is to shed some light on the validity of suh a

dihotomyforalllassesofgraphsusingtheVC-dimensionofthelassofgraphs.

VC-dimension. LetH= (V,E)^beahypergraph. AsubsetX⊆V ^{ofvertiesisshatterediffor}

everysubsetS^ofX^{,thereissomehyperedge}e^suhthate∩X =S^{. The}VC-dimensionofH^isthe

sizeofthelargestshatteredsetofH^{. Wedenethe}VC-dimensionofagraphastheVC-dimension of thelosedneighbourhood hypergraphof G^{(vertiesare thevertiesof}G^{and hyperedges are}

thelosedneighbourhoodsofvertiesofG^{),alassialwaytodenethe}VC-dimensionofagraph (see[1,6℄).

By a shattered set of a graph G^{, we mean a shattered set of the hypergraph of the losed}

neighbourhoods of G^{. The} VC-dimension of a lass of graphs C^{, denoted by} dim(C)^{, is the}

maximumoftheVC-dimensionofthegraphsoverC^{. Ifitisunbounded,wesaythat}C^hasinnite

VC-dimension.

Dihotomy for lower bounds. First we will prove in Setion 2 that there is indeed suh a

dihotomyontheminimumsizeofidentifyingodes: itisalwayseitherlogarithmiorpolynomial,

where the exponentof the polynomial depends on the VC-dimension of the lass of graphs. In

partiular,ourtheoremprovidesnewlowerboundsforgraphsofgirthatleast5^{,hordalbipartite}

graphs,unitdiskgraphsandundiretedpathgraphs. Moreover,theseboundsaretightforinterval

graphsandgraphs ofgirthatleast5^.

Graphlass VCdim IC-lower bound IC-approx

Girth≥5 2 Θ(n¹²)^{(opt,new) open}

Interval 2 Θ(n¹²)^{(opt) 6(Thm. 5.3 )}

Chordalbipartite 3 Ω(n¹³)^{(new) open}

Unitdisk 3 Ω(n¹³)^{(new) open}

C4^{-freebipartite 2} Θ(n¹²)^{(opt,new) no}clog(n)^{-approx(Thm. 4.3 )}

Undiretedpath 3 Ω(n¹³)^{(new) open}

Table2

Overviewoftheresultsobtainedinthispaper.

Approximation hardness. We then try to extend this dihotomy result for onstant fator

approximations. First,weshowinSetion3thatMinIdCodeislog^{-APX-hardforanyhereditary}

lasswith alogarithmilower bound. Theproofessentiallyonsistsin provingthat a hereditary

lass with innite VC-dimension ontains one of these three lasses, for whih Min Id Code

has been shown to belog^{-APX-hard [14℄: the bipartite graphs, the} o-bipartite graphs, or the split graphs. Unfortunately, the dihotomy does notextend to approximation sineweshow in

Setion 4 thatC4^{-freebipartite graphshavea polynomial lower bound onthesize of}identifying odes but Min Id Code is not approximable to within a fator clogn ^{for some} c > 0 ^(under

some omplexityassumption) in thislass. Thus, a onstant fatorapproximationis notalways

possiblein theseond ase.

Approximationalgorithm. Finally,inSetion5 ,weonludethepaperwithsomepositivere-

sultwhenthelowerboundispolynomialbyprovingthatthereexistsa6-approximationalgorithm forintervalgraphs,a problemleftopenin [14 ℄.

Theresultsobtained inthispaperaredetailed inTable2.

2. Dihotomy for lower bound. Most of the results using VC-dimension onsist in ob-

tainingupperbounds. However,in thelastfewyears,severalinterestinglower boundshavebeen

obtainedusingVC-dimension,forinstaneingametheory(e.g.[10 ,28 ℄). Alltheseproofsonsist

inanappliationofalemma,duetoSauer[30 ℄andShellah[32 ℄,oroneofitsvariants. Ourresult

hasthesameavoursineweusethislemmatoprovethat thesizeofanidentifyingodeannot

betoosmalliftheVC-dimensionisbounded. Thetrae ofa setX ^onY ^isX∩Y^{. Byextension,}

the traeof avertexx^onY ^istheintersetionofN[x]^withY^.

Lemma 2.1 (Sauer'slemma [30, 32 ℄). Let H= (V,E)^{be an hypergraph of} VC-dimension d^.

Foreveryset X⊆V^{,the numberof (distint)traesof} E ^on X ^isatmost

d

X

i=0

|X| i

≤ |X|^d+ 1.

Letusnowprovethemainresultofthissetion.

Theorem 2.2. Forevery hereditary lassof graphs C^,either

1. for every k ∈ N, there exists a graph Gk ∈ C ^{with more than} 2^k −1 ^{verties and an}

identifyingodeofsize2k^,or

2. there existsε >0 ^{suhthat no twin-free graph} G∈ C ^with n ^{vertieshas an} identifying odeofsizesmallerthann^ε.

Proof. Let C ^{bean hereditary lass of graphs. Thelass} C ^{either has nite or innite VC-}

dimension. First,supposethatC^hasinniteVC-dimension. WewillshowthatC^{satisestherst}

onlusion. Bydenition ofinniteVC-dimension,thereisagraph Hk ∈ C^withVC-dimensionk

x

x

x

Fig.1.Thesetx1, x2, x3 ^{isshatteredinthis}

hordalbipartitegraph.

Fig. 2. Aset of three verties shatteredby

disksintheplane.

foreahk^{. Sothere existsa set ofverties}X ^{of size}k ^ofHk ^{whihis shattered. Let}Y ^{bea set}

of 2^k −1 ^{verties whose losed} neighbourhoods haveall possible traes on X ^{exept the empty}

set, meaning that for every X^′ ⊆X^{, adda vertex} y ⁱⁿ Y ^{suh that} N[y]∩X =X^{′. Choose}Y

so that |X∩Y| ^{ismaximized. Let}Gk =Hk[X∪Y]^{. The graph}Gk ^{hasat least} 2^k−1 ^verties

sine|Y|= 2^k−1^{. Byhoieof}Y^,X ^dominatesX∪Y ^andX distinguisheseverypairofverties ofY^{. By maximalityof} |X∩Y|^, X ^alsodistinguishes every vertexin X ^{from every vertex in}Y

(otherwiseavertexofY ^{wouldhavethesame}neighbourhoodinX ^asavertexx∈X ^andthusan

bereplaedbyx^, ontraditingthemaximalityof |X∩Y|^{). For eah} x∈X^{, thevertex}yx ∈Y

whoselosedneighbourhoodintersetsX ^inexatly{x}distinguishesx^{fromallvertiesin}X−x^.

SoX∪ {yx|x∈X}^isanidentifyingodeofsizeatmost2k^,asrequired.

Now suppose that the VC-dimension of C ^{is bounded by} d^{. For any} identifying ode C ^of

a twin-free graph G ∈ C^{, the traes of verties of} G^on C ^{are dierent. Hene, by Lemma 2.1,} n≤Pd

i=0 |C| i

≤ |C|^d+ 1^{. Therefore,}|C| ≥(n−1)^{1d,provingthat}C ^{satises theseondlaim.}

Theproofgives in fat the lower bound γ^ID(G)∈ Ω(n^dim1(C)) ^{for theseond item. So ifwe}

an boundthe VC-dimensionof thelass, then weimmediately obtain lower bounds onthesize

ofidentifyingodes. Lemma 2.3providessuh bounds forseveral lassesofgraphs.

Letusgivesome denitions. Thegirth ofagraphisthelengthofashortestyle. Ahordal

bipartite graph is abipartite graph withoutinduedyleof lengthatleast 6^{. A unitdisk graph}

is agraph of intersetion ofunitdisks in theplane. Aninterval graph is a graph ofintersetion

of segmentson a line. Anundireted path graph is a graph of vertex-intersetion of paths in an

undiretedtree(i.e. twovertiesareadjaentiftheirorrespondingpathshaveatleastonevertex

inommon).

Lemma 2.3. The following upperboundsholdandaretight:

• ^The VC-dimension of graphsof girth atleast5^isatmost 2^.

• ^The VC-dimension of hordal bipartite graphs isatmost 3^.

• ^The VC-dimension of unitdiskgraphsisatmost 3^.

• ^The VC-dimension of intervalgraphsisatmost2^.

• ^The VC-dimension of undiretedpathgraphsisatmost3^.

Proof.

• ^LetG^{beagraphofgirthatleast}5^{. Assumeby}ontraditionthataset{x1, x2, x3}^ofthree

vertiesisshattered. Sinethegirthisatleast5^,x1x2x3 ^{isnotalique. Wemayassume}

withoutlossofgeneralitythatx1^andx2^{arenotadjaent. Sine}{x1, x2, x3}^isshattered,

there is a vertex y1 ^{adjaent to both} x1 ^and x2 ^{and not} x3 ^{(one losed} neighbourhood must have trae {x1, x2} ^on {x1, x2, x3}^{) and a vertex} y2 ^{adjaent to} {x1, x2, x3} ^(one

losedneighbourhoodmusthavetrae{x1, x2, x3}^{). Notethatboth}y1^andy2^aredistint

from x1 ^andx2 ^sinex1 ^andx2 ^{arenotadjaent. Moreover} y1 ^andy2 ^{aredistint sine}

they do not have the same neighbourhood in {x1, x2, x3}^{. So} x1y1x2y2x1 ^{is a yle of}

length4^,a ontraditionwiththegirthassumption.

Thisboundistight,forinstane withthepathonsixverties.

• ^LetG= (A∪B, E)^{beahordalbipartitegraph.Assumeby}ontraditionthat{x1, x2, x3, x4}

P

P

P

Fig.3. PathsP1, P2, P3 ^{areshatteredbytheeightpointswhiharepathsoflength}0^.

isashatteredsetoffourverties. Sinethereisavertexwhoselosedneighbourhoodon-

tains the whole set of verties, it means that at least three verties, say x1, x2, x3 ^are

on the same side of the bipartite graph. Sine a subset of a shattered set is shattered,

{x1, x2, x3} ^{is shattered. Thus there is a vertex inident to} x1, x2 ^{and not}x3^{, a vertex}

inidentto x1, x3 ^{and not}x2^{,and a vertexinidentto} x2, x3 ^{and not}x1^{. Itprovidesan}

induedyleoflength6^,aontradition.

Moreover theboundistight,seeFigure1.

• ^LetG ^{bea unit diskgraph. Letus rephrase theadjaenyand shatteringonditions in}

this lass: letx1 ^and x2 ^{beany two verties ofa unit diskgraph anddenote by} c1 ^and

c2 ^{their respetiveenters in a} representation of the unitdisk graph in the plane. The verties x1 ^and x2 ^{areadjaentifand onlyif}c1 ^and c2 ^{are atdistane at most}2^{. Thus}

ifa set ofunitdisksis shatteredthenforeverysubsetof enters, thereexists a pointat

distane atmost2 ^{fromthese enters andmore than2 fromtheothers. Inotherwords,}

thereexist pointsin allpossible intersetionsofballsofradius2^.

A lassialresult ensuresthat theVC-dimension of ahypergraphwhose hyperedges an

berepresentedasa setofdisksin theplane(andvertiesaspointsoftheplane)hasVC-

dimensionat most3^{(see[26 ℄forinstane). Thusunitdiskgraphshave}VC-dimensionat most3^{,andtheboundanbereahed(seeFigure2).}

• ^Let G ^{be an interval graph. Assume by} ontradition that there is a shattered set

{I1, I2, I3} ^of G^{. Assume that} I1 ^{starts before} I2 ^{and that} I2 ^{starts before} I3^{. Sine}

there is an intervalJ interseting bothI1 ^and I3 ^{but not} I2^, J ^{must startafter} I2 ^and

thusI1^ontainsI2^{. Thenthereisnointerval}interseting I2 ^{but not}I1^,aontradition.

ThusintervalgraphshaveVC-dimensionatmost2^{,andthebound isagainreahedwith}

thepathonsixverties.

• ^LetP ={P1, P2, P3, P4} ^{bea shattered set offour pathsof a tree}T^{. Assume rstthat} P2, P3, P4^allintersetP1^{andonsidertherestritionof}T^toP1^{,whihisinfataninterval}

graph. ToensureallpossibleintersetionswithP1^,theset{P2, P3, P4}^{isa shatteredset}

ofsizethree inanintervalgraph,a ontradition.

Thusat leastonepath, sayP2^{, doesnotinterset} P1 ^{and liesina onnetedomponent}

C ^oftheforestF =T \P1^{. If}P3 ^{doesnotinterset}C^{,thenthere isnopath}interseting bothP2 ^andP3^{but not}P1^{. Thus}P3^intersets C^{. Ifmoreover}P3 ^intersetsP1^{, thenno}

pathan interset bothP1 ^andP2 ^butnotP3^{. Thus} P3 ^{isalso inludedin} C^{. Let}P ^be

a pathintersetingP1^,P2 ^andP3^{. Assume rstthat} P ^{intersetsthethree pathsin the}

orderP1^,P2^andP3^(theaseP1^,P3^,P2^{)isthesame. Thennopathaninterset}P1^and

P3 ^withoutinterseting P2^{. Assume now that} P ^{intersets thethree paths in theorder} P2^,P1^,P3^{. Similarly,nopathaninterset}P2^andP3^withoutintersetingP1^{. Henethe}

pathP ^annotexist,aontradition. Finallytheboundof3^{anbereahed,asshownin}

Figure3 .

Lemma2.3andTheorem2.2implynewlowerboundsformanylasses: Ω(n¹²)^{forgraphswith}

girthatleast5^,Ω(n¹³)^{forhordalbipartitegraphs,}Ω(n¹³)^{forunit-diskgraphs,}Ω(n¹²)^forinterval

graphs,Ω(n¹³)^forpermutation graphs,andΩ(n¹³)^{forundiretedpathgraphs.}

Theexponentgivenby Theorem2.2 is sharpfor several lassesof graphs. Indeed, Fouaud