Sur la Conjecture des Jeux Uniques

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Sur la Conjecture des Jeux Uniques

Stéphane Pérennes

To cite this version:

Stéphane Pérennes. Sur la Conjecture des Jeux Uniques. [Research Report] RR-6691, INRIA. 2008.

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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Sur la Conjecture des Jeux Uniques

Stéphane Pérennes — Ignasi Sau

N° 6691

Unité de recherche INRIA Sophia Antipolis

2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)

Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65

Sur la Conje ture des Jeux Uniques

∗

Stéphane Pérennes

†

, IgnasiSau

‡§

ThèmeCOMSystèmes ommuni ants

ProjetMASCOTTE

Rapportdere her he n°6691September20088pages

Résumé: Laplupart desproblèmesd'optimisation ombinatoiresontNP-di iles,

'est-à-direqu'ils ne peuvent être résolus en temps polynomialque si les lassesP et NP sont

identiques.Pour esproblèmesonpeutespérersoittrouverdesalgorithmesd'approximation,

soitprouverqu'ilsnepeuventpasêtreapproximésdemanièree a e.

En2002S.KhotformulalaConje turedesJeuxUniques (UGC),quigéneraliselethéorème

PCP et impliquerait d'importants résultats d'innaproximabilité pour plusieurs problèmes

d'optimisation ombinatoire (parexemple Max Cut ouVertex Cover). Intuitivement,

laUGCditque,pourune ertaine lassedejeux, appelésuniques,ilestNP-durdedé ider

sil'onpeuttrouverunesolutionpro he del'optimale,ousitouteslessolutionssontloin de

l'optimale.

Cette onje tureestdevenueunproblèmeouvertdesplusimportantsdanslathéoriedela

omplexitéetdel'approximation.

Dans et arti le nous étudions un problème très relié à la UGC: Max-E

2

-Lin2 dans les graphesbipartis.DansMax-E

2

-Lin2onaungraphe

G

ayantdeuxtyped'arêtes,requérant soitlamêmesoitdiérente ouleurpoursesextrémités.Lebutestde2- olorerlessommets

de

G

enmaximisantlenombred'arêtessatisfaites.Nousprouvonsque eproblèmeest APX- ompletdanslesgraphesbipartiset,enutilisantleThéorème de RépétitionParalèlle,nous

dis utonsles onséquen esde e résultatdansle adredesjeuxuniques etlaUGC.

Mots- lés : ComputationalComplexity, Unique Games,ParallelRepetition, Hardnessof

Approximation.

∗

This workhas been partially supported byEuropean proje t IST FETAEOLUS, PACAregion of

Fran e,MinisteriodeEdu a iónyCien iaofSpain,EuropeanRegionalDevelopmentFundunderproje t TEC2005-03575,CatalanResear h Coun ilunderproje t2005SGR00256 andCOSTa tion293GRAAL, andhasbeendoneinthe ontextofthe r CorsowithFran eTele om.

†

Mas otte joint proje t - INRIA/CNRS-I3S/UNSA, Sophia-Antipolis, FRANCE. Stephane.Perennessophia. inri a.f r

‡

Mas otte joint proje t - INRIA/CNRS-I3S/UNSA, Sophia-Antipolis, FRANCE. Ignasi.Sausophia.inria.f r

§

GraphTheoryandCombinatori sgroupatAppliedMathemati sIVDepartmentofUPC,Bar elona, SPAIN.

A Note on Max-E

k

-Lin2 and the Unique Games Conje ture

¶

Abstra t: Most ombinatorial optimization problems are NP-hard, i.e. they annot be

solvedinpolynomialtimeunlessP

=

NP.Twotypi alapproa hesto dealwiththese prob-lemsareeithertodeviseapproximationalgorithmswithgoodperforman e,ortoprovethat

su halgorithms annotexist.In2002S.KhotstatedtheUniqueGamesConje ture (UGC),

whi hgeneralizesthePCPTheorem.TheUGCwouldimplysigni anthardnessresultsfor

severaloptimization problems(e.g., MaxCut orVertex Cover). Looselyspeaking,the

UGC statesthat for a lass of games ( alled unique) in whi h theoptimal solution takes

valuesbetween0and1,it isNP-hardtode idewhether theoptimalis lose to0or lose

to1.

This onje turehasbe omeoneof themostoutstanding openproblemsin omplexityand

approximation theory. In this arti le we study a problem whi h turns out to be losely

related to the UGC: Max-E

2

-Lin2 in bipartite graphs. The input of Max-E

2

-Lin2 is a graph

G

with twotypesofedges.Namely,ea hedgerequires itsend-verti esto be olored witheither the same olorordierent olors.Theobje tiveis to 2- olor theverti esof

G

maximizing thenumber ofsatised edges. Theproblem is known to beAPX- omplete in

generalgraphs.Inthisarti leweprovethattheproblemremainsAPX- ompleteinbipartite

graphsand, using the Parallel Repetition Theorem, we dis uss the onsequen es that this

result ouldhaveintheframeworkof uniquegamesandtheUGC.

Key-words: ComputationalComplexity,Unique Games,ParallelRepetition,Hardnessof

1 Introdu tion

A unique game onsists of an undire ted onne ted graph

G = (V, E)

, a olor set

C

, and for ea h edge

(i, j)

with

i < j

apermutation

π

i,j

: C → C

. A oloring of the graph

c : V → C

fullls (or satises) an edge

(i, j)

if

π

i,j

(c(i)) = c(j)

. Trevisan gives a ni e introdu tionto uniquegames in [10℄.If a oloringfullls allthe edges, then knowingthe

oloratonevertexuniquelydeterminesalltheother olors.Thatisthereasonwhysu h a

gameis alledunique.One ane ientlydeterminewhethersu ha oloringexistsbytrying

all possible olors at onenode to see if anyof the resulting oloring fullls all the edges.

Howeveritmightbedi ulttodeterminewhetherone anfulllsomelargefra tionofthe

edges.In2002, SubhashKhot denedtheUniqueGames Conje ture (UGC forshort):

Conje ture 1.1 (UniqueGames Conje ture [4℄) For every onstant

δ > 0

there isa xednite olor lass

C

su hthatitis NP-hardtodistinguishthe following 2 asesfor any uniquegame with olor lass

C

:

1. Thereissome oloring thatfulllsatleasta

(1 − δ)

-fra tionof the edges. 2. Every oloringfulllsatmosta

δ

-fra tionof the edges.

Sin eit wasrstformulatedin 2002,theUGChasbe omeoneof themost hallenging

openproblemsin omputational omplexity[5,6,8,10℄.TheUGCis astrengtheningofthe

Probabilisti ally Che kable Proof (PCP) Theorem. If the UGC were true, it would have

important onsequen esfor the theory of omplexity and approximation.For instan e, it

wouldimplythatanyimprovementinapproximatingMaxCutbelow0.878567wouldfor e

P

=

NP[6℄,andthatVertex Cover wouldbehardtoapproximatewithin 2-

ε

[7℄. In this arti le we study a problem whi h seems to be losely related to a parti ular

aseoftheUGC. Namely, in Se tion2weprovethat Max-E

2

-Lin2in bipartitegraphsis APX- omplete, bya redu tionfrom

3

-Max-Cut. InSe tion 3wedis uss the relation of this result with the UGC,in order to understand better the UGC,or even to providean

intermediatesteptowardsahypotheti proofoftheUGC.

2 Max-E

2

-Lin2 in Bipartite Graphs is APX- omplete

Inthisse tionwefo usonthefollowing lassi alNP- ompleteproblem:

Max-E

k

-Lin2:

Input:asetoflinearequationsmodulo2withexa tly

k

variablesperequation. Output:anassignmentofthevariablesmaximizingthenumberofequationssatised.

The asewhen

k = 2

anbeseenin anaturalwayasagraphoptimizationproblem as follows: ea h variable orrespondsto avertex,and ea h linearequation (involvingexa tly

2 variables) orresponds to an edge. There are two typesof edges, namely those that

re-quire theirendpointsto havethesame value (either 0or 1), and those that requiretheir

endpoints to have dierent value. Inwhat follows, we allsu h edges of type

S

and type

D

,respe tively. Letalso

|S|

and

|D|

bethenumberofedges oftype

S

and

D

oftheinput graph

G

, respe tively. In this ontext, an optimal solution is a 2- oloring of the verti es

of

G

maximizingthenumberofsatisededges.Intheedge-weightedversion,theobje tive fun tionto bemaximizedisthesumoftheweightsofthesatisededges.

Karpinski proved in [3℄ that Max-E

2

-Lin2 is APX- omplete in general graphs. It is natural to ask what happens to omplexity when we restri t the input graph to being

bipartite(Max-E

2

-Lin2-Bipforshort).WeproveinTheorem2.1thatMax-E

2

-Lin2-Bip remainsAPX- omplete.Themainreasonwhywestudythisproblem,besidesthefa t that

thehardnessresultisinterestingbyitself,isits loserelationwiththeUGC,asweshallsee

inSe tion 3.BeforegoingthroughtheproofofTheorem2.1,weneedto introdu eanother

lassi alNP- ompleteproblem:

d

-Max-Cut:

Input:unundire tedgraph

G = (V, E)

ofdegreebounded by

d

.

Output : apartition of

V

into twogroups so as to maximize the number of edges withexa tlyoneendpointin ea hgroup.

ObservethatsolvingMax-Cutinagraph

G

isequivalenttondingabipartitesubgraph withmaximumnumberofedges,sin eanybipartitesubgraph

H ⊆ G

withmaximumnumber ofedges anbetransformedintoanother(notne essarilyindu ed)bipartitesubgraph

H

′

_⊆

G

ontaining

H

su h that

|E(H

′

_{)| ≥ |E(H)|}

and

V (H

′

_{) = V (G)}

.

Karpinskiprovedin[3℄that3-Max-CutisAPX- omplete,providinganapproximation

algorithmwithapproximationratio1.0858,togetherwithahardnesslowerboundof1.003.

Theorem2.1 Max-E

2

-Lin2-Bipis APX- omplete.

Proof: ToseethatMax-E

2

-Lin2-BipbelongstoAPX,itiseasytosatisfy

max{|S|, |D|}p

edgesof theinput bipartitegraph

G = (A ∪ B, E)

just byrst oloring alltheverti esof

G

withthesame olor(fullling

|S|

edges),then oloringtheverti esin

A

with olor0and theverti esin

B

with olor1(fullling

|D|

edges),andtakingthebestsolution.Thisnaïve algorithm learlyprovidesa2-approximation.

We provethat Max-E

2

-Lin2-Bip is APX-hardwith agap-preservingredu tion from 3-Max-Cut, provedtobeAPX-hardin[3℄. Werstdealwiththe edge-weightedversion,

andthenwedes ribehowtoadapttheredu tiontotheunweighted ase.

Givenaninputgraph

G = (V, E)

of 3-Max-Cut withmaximumdegreeat most3,we onstru t an instan e

G

′

of Max-E

2

-Lin2-Bip in the following way : we put two opies

V

1

and

V

2

of the vertex set

V

, i.e. there are twoverti es

u

1

∈ V

1

and

u

2

∈ V

2

for ea h vertex

u ∈ V

. (Withslightabuseofnotation,wewill supposehen eforththat toea hpair

{u

1

, v

2

} ∈ V

1

× V

2

orresponds without ambiguity a pair

{u, v} ∈ V × V

, with possibly

u = v

.)Thenweadd anedgeoftype

D

withweight1between

u

1

∈ V

1

and

v

2

∈ V

2

ifand onlyif

(u, v) ∈ E

,andanedgeoftype

S

withweight

4

between

u

1

∈ V

1

and

v

2

∈ V

2

ifand only if

u = v

. This ompletes the onstru tion of

G

′

_{= (V}

1

∪ V

2

, E

′

)

, whi h is illustrated withasmallexampleinFig.1a.Let

n = |V (G)|

.

Claim1 Anyoptimalsolutionof Max-E

2

-Lin2-Bipin

G

′

ontainsthe

n

edgesof type

S

. Letussee thatgivenanysolution

H

of Max-E

2

-Lin2-Bipin

G

′

,we antransform

H

into abetter(in termsof the weightof the satisededges)solution

H

′

ontainingthe

n

edges

u

v

w

z

u

v

w

z

1

1

1

1

u

v

w

z

2

2

2

2

G

G'

4

4

4

4

u

1

1

u

1

2

u

1

3

u

4

₁

v

₁

4

v

₁

3

v

2

₁

v

1

1

u

2

1

u

2

2

u

2

3

u

₂

4

v

₂

4

v

₂

3

v

₂

2

v

2

1

u

v

G

_G'

b)

a)

Fig.1Redu tionin theproofofTheorem 2.1: a)fortheweighted ase;and b)forthe

unweighted ase.Thethi kedgesareoftype

S

,theothersbeingoftype

D

.Ifnoweightis displayedbesideanedge,itisassumedtobe1.

of type

S

. Indeed,iffor avertex

u

the edge

(u

1

, u

2

)

is notsatisedin

H

, then

u

1

and

u

2

havedierent olorin

H

.Sin ethedegreeof

u

in

G

isatmost3,ifwe hangethe olorof

u

1

(let

H

′

bethisnewsubgraph)thenatmost3edgeswithweight1be omeunsatisedin

H

′

,but theedge

(u

1

, u

2

)

withweight

4

issatisedin

H

′

. Thus,theweightofthe satised

edgesof

H

′

isstri tlygreaterthantheweightofthoseof

H

.

Let

OP T

G

be the optimal value of 3-Max-Cut in

G

, and let

OP T

G

′

be the optimal valueoftheweightedversionof Max-E

2

-Lin2-Bipin

G

′

.

Claim2

OP T

G

′

= 2 · OP T

G

+ 4n

. ByClaim1,inanoptimalsolution

H ⊆ G

′

,forea hvertex

u ∈ V (G)

,bothverti es

u

1

and

u

2

are olored withthe same olor.So ifthe edge

(u

1

, v

2

)

is satisedin

H

, then theedge

(u

2

, v

1

)

is alsosatised.Therefore,

OP T

G

′

is a hieved by a2- oloring ofthe verti es of

G

maximizingthenumberofedgeswithexa tlyoneendpointinea h olor lass.Thisproblem

isexa tly3-Max-Cut in

G

,sothe laimfollows.

ByClaim2,itis learthattheexisten eofaPTASforMax-E

2

-Lin2-Bipwouldimply theexisten eofaPTASfor3-Max-Cut, whi h isa ontradi tion.

Forthe unweighted ase, we modify the onstru tion of

G

′

in the following way : for

ea h vertex

u ∈ V (G)

, weput 8 opies

u

1

1

, . . . , u

4

1

∈ V

1

and

u

1

2

, . . . , u

4

2

∈ V

2

. Thenweadd a omplete bipartitegraphwith edges oftype

S

betweentheverti es

u

1

1

, . . . , u

4

1

∈ V

1

and

u

1

2

, . . . , u

4

2

∈ V

2

. Finally, we add an edge of type

D

between

u

4

1

∈ V

1

and

v

4

2

∈ V

2

if and onlyif

(u, v) ∈ E

(seeFig.1b).On eagain,anyoptimalsolutionfor

G

′

ontainsallthe

16n

edgesoftype

S

,and

OP T

G

′

= 2 · OP T

G

+ 16n

.Thetheorem follows.

2

3 Relation with the Unique Games Conje ture

Inthis se tion wedis uss therelation ofTheorem 2.1 withtheUGC. Firstweneed to

introdu ethe Parallel Repetition Theorem, proved byRaz in [9℄and further simplied by

Holensteinin [2℄. This resultstates that aparallel repetition of any two-prover one-round

proof system (it an be seenasa uniquegame in abipartite graph,see [1℄) de reasesthe

probabilityoferroratanexponentialrate.

Max-E

2

-Lin2-Bip anbenaturallythoughtofasauniquegame

G

with

|C| = 2

olors, the permutation for ea h edge being either the identity or a transposition. The parallel

repetition of

k

times

G

orrespondsto aunique gamewith

|C| = 2

k

olors.Indeed, letus

des ribethegamewhenwerepeat

G

on e.Giventheinputgraph

G

of Max-E

2

-Lin2-Bip, we onsiderthe dire t produ t

G × G

, and then

4

types ofedges anappear,namely

SS

,

SD

,

DS

,and

DD

.Wehavethat

C =

Z

2

2

and

|C| = 4

(moregenerally,ifwerepeat

k

times

G

wehavethat

C =

Z

k

2

and

|C| = 2

k

).Thisgameisabelian inthesensethat,forinstan e,an

edge

((a, x), (b, y)) ∈ E(G × G)

oftype

SD

issatisedifandonlyif

c(a) + c(b) ≡ 0 (mod 2)

and

_{c(x) + c(y) ≡ 1 (mod 2)}

. Observethat in this parti ular game all the bije tions are translations.

Usingthenotationof[2,9℄,let

v < 1

betheoptimalvalueof

G

. This onstant

v

anbe seenasthemaximumprobabilitythat bothplayers anwin, ea h player orresponding to

onestable set of the bipartitegraph

G

. This probability

v

orrespondsto the per entage ofsatisededgesofanoptimalsolutionin

G

. TheParallelRepetition Theoremstatesthat when we repeat

k

times

G

in parallel,then the probability that bothplayers an win all gamessimultaneouslyisatmost

v

k

,where

v < 1

isa onstantdependingonlyon

v

. Ontheonehand,theUGCstatesthatforeveryxed onstant

δ > 0

,thereexistsa olor lass

C

su h that thegap (givenby ases1and 2in Conje ture1.1) foranyuniquegame with olor lass

C

is

[δ, 1 − δ]

.

Ontheotherhand,Theorem2.1statesthatthereexistsa onstant

0 < α < 1

su hthat itisimpossible(unlessP

=

NP)tondinpolynomialtimeasolutionbetter than

(1 − α)v

, intermsoftheper entageofsatisededges.ThisAPX-hardnessresult anbeseenasagap

[(1 − α)v, v]

forthegame

G

, inthe sensethat in thisintervalwehaveno knowledge about theoptimalsolution.

Inorderto get losetotheUGC,weneedtobeabletosaysomethingmoreaboutthis

gapfor

G

.Ifonemanagedtoprovethatthegapforthegame

G

orrespondingtoMax-E

2

-Lin2-Bip were, for instan e,

[1 − 1/k

2

_{, v}

1/k

_]

, then theParallelRepetition Theoremwould

implythat thegap ofrepeating

k

times

G

would be,asymptoti ally,

[1 − 1/k, v]

. Sin efor anyxed onstant

δ > 0

thereexistsaninteger

k

su hthat

1/k < δ

,theinterval

[1 − 1/k, v]

wouldbe ontainedin

[1 − δ, 1]

,andtheupper partofthegapgivenbytheUGCwouldbe overed.Finally,thesamekindofargument ouldbeadaptedto overthelower partofthe

gap.

Let

ˆ

v

be a onstantsu hthat

1 − 1/k

2

_{< ˆ}

_{v < 1}

.Summarizing,the importantquestion

for

G

isthefollowing:Providedthat weknowthat aproportionat least

(1 − 1/k

2

₎

ofthe

edgesof

G

an besatised,is itNP-hard to satisfyaproportion

ˆ

v

of theedges? Inother words,theproblemiswhetherMax-E

2

-Lin2-Bip anbee ientlyapproximatedwhenone

knowsthatin anoptimalsolutionalmost alledgesaresatised.Equivalently, isitpossible

tondabipartitionofagraphwhenitisalmost bipartite?

TheUGCforthis lassofgameswouldbetrueifwewereabletohavemoreinformation

on erning the two ases of Max-E

2

-Lin2-Bip, i.e. the ase where any solution satises very fewedges, and the asewhere there exists asolutionsatisfying almost alledges. We

haveseenin theproofofTheorem2.1that, modulo anadditivefa tor,Max-E

2

-Lin2-Bip looks like 3-Max-Cut, soabetterunderstanding of Max-Cut ouldalso provide

impor-tantinsightsinto theUGC.

In on lusion, we showed that the problem of Max-E

2

-Lin2 is APX- omplete in bi-partite graphs,and wesaw that the parallel repetition of the game orresponding to this

problemnaturallyleadsto uniquegames.Due to thisstrong relation,it would bevery

in-terestingto understand thehardnessof Max-E

2

-Lin2better, in orderto shed somelight ontheUnique GamesConje ture on erninguniquegames.

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4, rue Jacques Monod - 91893 ORSAY Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique

615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)

Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38334 Montbonnot Saint-Ismier (France)

Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

Éditeur

INRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)

http://www.inria.fr