An exact method for graph coloring

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An exact method for graph coloring

Corinne Lucet, Florence Mendes, Aziz Moukrim

To cite this version:

Corinne Lucet, Florence Mendes, Aziz Moukrim. An exact method for graph coloring. Computers and Operations Research, Elsevier, 2006, 33 (8), pp.2189-2207. �hal-00783637�

An exat method for graph oloring

C. Luet, F. Mendes

LaRIA EA 2083, 5 rue du Moulin Neuf 80000 Amiens - Frane

(Corinne.Luet, Florene.Mendes)laria.u-piardie.fr

A. Moukrim

HeuDiaSyC UMR CNRS 6599 UTC, BP 20529 60205Compiegne - Frane

Aziz.Moukrimhds.ut.fr

November 15,2004

Abstrat

Weareinterestedinthegrapholoringproblem. Weproposeanexat

methodbasedonalinear-deompositionofthegraph. Theomplexityof

thismethodisexponentialaordingtothelinearwidthoftheentrygraph,

butlinearaording to itsnumberofverties. Wepresent someexperi-

mentsperformedonliteratureinstanes,amongwhihCOLOR02library

instanes. Ourmethodis usefullto solvemorequiklythanotherexat

algorithmsinstaneswithsmalllinearwidth,suhasmug graphs. More-

over,ouralgorithmsarethersttoourknowledgetosolvetheCOLOR02

instane4-Inser3 withanexatmethod.

Keywords: grapholoring,exatmethod,linearwidth,linear-deomposition.

1 Introdution

The notions of tree-deomposition and path-deomposition have been intro-

duedbyRobertsonandSeymour[23℄. Thedeompositionmethodwepropose

hereisstronglyrelatedtothese notions,whihhavebeenstudied inpartiular

byBodlaendertosolvesomeNP-hardproblems[1℄.

Ourapproahisamethod basedonsuessivedeompositionsoftherepre-

sentativegraphprovidingsuessiveresolvedsubgraphsandtheirorresponding

withthenanialsupportofConseilRegionaldePiardieandFSE

grapharerepresentedbythedierentstatesof theboundarysetverties. The

numberofstatesto enumerategrowsexponentiallywiththesizeofthebound-

ary set. Its maximum size, for an optimal vertex numbering, orresponds to

thelinearwidth of thegraph. The main advantageof this method is that the

exponential fatorof its omplexity doesnot depend onthe size of the graph

but only on its linearwidth. This tehnique has been implemented eÆiently

byCarlier, Luetand Manouvrier to solve various NP-hardproblems suh as

network reliabilityorminimalSteinertreeomputation[4,18,19℄.

We apply the deomposition method to oneof the most studied problems

of ombinatorial optimization: the graph oloring problem. It onstitutes a

entralproblem in alot ofappliationssuhasshooltimetabling,sheduling,

orfrequenyassignment[5,6℄. Thegrapholoringproblemonsistsin oloring

the verties of a graph with a minimum number of olors, ensuring that two

adjaent verties do notreeivethe same olor. Various heuristi approahes

havebeen proposed forthis NP-hardproblem [12℄: greedy algorithms suh as

DSATUR [3℄, metaheuristis based on loal searh, tabu method, simulated

annealing,hybridalgorithms,et. (seeforexample [10,11,13,16,20,22,26℄).

To our knowledge, few exat methods are proposed to resolve this problem.

One of the most well-known exat algorithms is the exat branh-and-bound

algorithm implemented by Brelaz that uses DSATUR priniples [3℄. Impliit

enumerationstrategiesareusedin[17,25,27℄. MehrotraandTrik[21℄studied

alinearprogrammingformulationwhih is solvedbyusing olumn generation

tehniques. Morereently,MendezDiazandZabalapresentedabranh-and-ut

algorithm[8,9℄. HerrmannandHertzpresentedeÆientalgorithmsusedtond

edge-ritialand vertex-ritial subgraphsthat havesame hromati numbers

as initial graphs but are easier to solve [15℄. Desrosiers, Galinier and Hertz

proposeddierentalgorithmstodetettheseritialsubgraphs[7℄. Theymade

experiments on random graphs and on dierent types of benhmark graphs.

Theirmethod isveryeÆientonseveralinstanesfamilies.

Ourpaperisorganizedasfollows. Insetion2,wedesribethedeomposi-

tionmethodandintroduetheneessarynotions. Insetion3,wedevelopthe

implementationofthemethod. Wepresentanexatalgorithmwhihenablesus

tosolveeÆientlylargeinstaneswhoselinearwidthisbounded. Computational

resultsobtainedonvariousinstanes arepresentedin setion4. Theyompare

withtwoexatmethods: abranh-and-utalgorithm[9℄andanalgorithmbased

onvertex-ritialsubgraphsdetetion[7℄. Finally,wegivesomeonlusionsand

disussabouttheperspetivesofthiswork.

Tointroduethekindofdeompositionthatweusetosolvethegrapholoring

problem,itisneessarytoreallsomegraphtheorydenitionsandthenotions

oftree-deompositionandpath-deomposition.

2.1 Preliminary denitions

AnundiretedgraphGisapair,G=(V;E),madeupofavertexsetV andan

edgesetEVV. AgraphGisonnetedifforallvertiesw;v2V(w6=v),

thereexistsapathfromwtov. Withoutlossofgenerality,thegraphsGwewill

onsiderinthefollowingofthispaperwillbeundiretedandonnetedgraphs.

A subgraph of G= (V;E), induedby W V, is agraph G(W)= (W;E

W )

suhthatE

W

=E\(WW). Atreeisasimpleundiretedgraph,T =(I;E

T ),

withoutyleandwithjE

T

j=jIj 1. A rootedtree isatreediretedfrom the

rootr tothe leaves. Iftheedge (p;v) belongs to arooted tree, pis thefather

ofv, andvisoneofthesons ofp.

2.2 Tree-deomposition

A tree-deomposition of G = (V;E) is apair (fX

i

=i 2 Ig;T = (I;E

T )) with

fX

i

=i2IgafamilyofsubsetsofV and T atreesuhthat:

{ S

i2I X

i

=V,

{for alledges(v;w)2 E, there exists asubsetX

i

;i2I; withv 2X

i and

w2X

i ,

{foralli;j;k2I,ifj isonthepathfrom itokin T thenX

i

\X

k X

j .

Thetreewidth ofatree-deomposition ismax

i2I (jX

i

j 1). Thetreewidth ofa

graphGistheminimumtreewidth overallpossibletree-deompositionsofG.

Thedeompositionmethodisasfollows. Givenatree-deompositionof the

graph,partialsolutionsarebuiltonthesubsetsX

i

andthenassoiatedtosolve

the onsidered problem. The deomposition method omputes the solutions

fromtheleavestotherootofthetreeT,byexaminatingallpartialsolutionson

everysubgraphG(X

i

). Thenumberofpartialsolutionsisexponentialaording

tothesizeofthesubgraphsX

i

. Thesepartial solutionsareomputedfrom the

solutions of X

f

, for all f belonging to the sons of i in T. Unlike a simple

enumerative method, this method allows one to fatorize partial solutions of

theX

i

sets into lasses. This fatorization provides aneÆientmethod if the

ardinalityofthesetsX

i

issmall,i.e. ifthetreewidthofthetree-deomposition

issuÆientlysmall.

onean omputethetreewidthin lineartime,omputingthetreewidth ofany

graphisaNP-ompleteproblem[24℄. Bodlaender[2℄givesforaonstantkan

algorithm in O(n) whih for a graph G solves the problem \is the treewidth

ofGat mostk?". Ifso,it determinesatree-deompositionwithtreewidth at

mostk. This algorithm based onlique searh and graphontration hasan

exponential omplexity with respet to k (O(n2 k

2

)). It annot be used in

pratie,evenfork=4.

2.3 Path-deomposition and linear-deomposition

Thedeompositionmethodthatwewilluseinthefollowingisbasedonaspeial

aseoftree-deomposition. Wewillonsideratreewithonlyoneleaf,thatisa

path.

Apath-deomposition (X

1

;:::;X

r

)ofagraphGisanorderedsequeneofsub-

setsofV suh that:

{ S

1ir X

i

=V,

{for alledges(v;w)2 E, there exists asubsetX

i

;1i r;with v 2X

i

andw2X

i ,

{foralli;j;k2f1;:::;rg ,ifijkthenX

i

\X

k X

j .

Thepathwidth ofapath-deompositionis max

1ir (jX

i

j 1). Thepathwidth

of a graph is the minimum pathwidth over all possible path-deompositions

of G. A vertex linear ordering of G is a bijetion N : V ! f1;:::;jVjg.

For morelarity, we denote k thevertex N 1

(k). Let F

i

=fj 2 V=9(j;l) 2

E j i < lg 8i2 f1;:::;jVjg. The linearwidth of a vertex linear ordering

N is F

max

(N) = max

i2V (jF

i

j). The linearwidth of G, written F

max (G), is

the minimum linearwidth over all possible vertex linear orderings of G. The

linearwidthofagraphequalsitspathwidth[19℄.

ComputingthepathwidthorthelinearwidthofanygraphisaNP-omplete

problem[24℄,similarlyasomputingthetreewidthofanygraph. Thetreewidth

ofagraphGissmallerorequaltoitspathwidth,andasaonsequenetheex-

ponentialfatorofatree-deomposition issmallerthan that ofapath-deom-

position. Nevertheless, implementing the deomposition method on a linear-

deompositioniseasierthanusingatree-deomposition. First,fromatehnial

point of view, several X

i

partial solutionsmay have to be stored in memory

when resolving a problem with atree-deomposition. It involvessome prob-

lems of memory storage and ombination of the X

i

when implementing the

algorithm. Moreover,reatingalinear-deomposition iseasier thanreatinga

solvethe grapholoringproblem with alinear-deomposition. Theresolution

method is then based ona sequential insertionof the verties, using a vertex

linearorderingpreviouslydetermined. Thiswill bedeveloped in thefollowing

setion.

3 Appliation to the graph oloring problem

Inthissetion,weproposeamethod whih useslinear-deompositionin order

tosolvethegrapholoringproblem.

3.1 Problem denition

Aoloring of agraphG=(V;E) is anassignment ofaolor (i)2I to eah

vertexsuh that(i)6=(j)foralledges(i;j)2E.

IftheardinalityofI isk,theoloringofGisalledak-oloring. Theminimum

valueof kfor whiha k-oloringispossibleis alled thehromati number of

G and is denoted (G). The graph oloring problem onsists in nding the

hromatinumberofagraph.

3.2 Linear deomposition priniple

ConsideragraphG =(V;E). LetN =jVj and M =jEj. The verties of G

are numberedaordingto alinear ordering N : V ! f1;:::;Ng. Let V

i be

subsetof V, madeof theverties numberedfrom 1to i. LetH

i

=(V

i

;E

i )be

thesubgraphofGinduedbyV

i . F

i

istheboundarysetofH

i

,i.e. thesubsetof

V

i

suhthatv2F

i

ifandonlyif9(v;w)2E andvi<w(seegure1). Let

H 0

i

=(V 0

i

;E 0

i

)bethesubgraphofGinduedbyV 0

i

=(V nV

i )[F

i

. Anykindof

relationbetweenthevertiesofH

i

andthose ofH 0

i

dependson thevertiesof

F

i .

The linear deomposition is a dynami method. During the oloring we

willonsiderN subgraphsH

1

;:::;H

N

andtheN orrespondingboundarysets

F

1

;:::;F

N

. Startingfrom avertexlinearordering, webuildatrstiterationa

subgraphH

1

whihontainsonlythevertex1,thenateahstepthenextvertex

anditsorrespondingedgesareadded,untilH

N

. Partialsolutionsofstepiare

builtfrom partialsolutionsofstepi 1.

AteahsubgraphH

i

orrespondsaboundaryset F

i

ontainingtheverties

ofH

i

whihhaveatleastoneneighborinH 0

i

. TheboundarysetF

i

isbuiltfrom

F

i 1

byaddingthevertexiandremovingthevertiesthathavenoneighborwith

4 15

2 8 10 16

5 14

1 7 11 17

6 13

12 18

G=(V;E)

3 9

4

2 8 10

5

1 7

6

15

8 10 16

14

7 11 17

13

12 18

H

10

H 0

10

Figure1: SubgraphH

10

of GanditsboundarysetF

10

=f7;8;10g

1

F

5

=f4;5g C(H

5

;1)=[45℄val(C(H

5

;1))=2

3 5

V

V B

V

B

2 4

1

F

5

=f4;5g C(H

5

;1)=[45℄val(C(H

5

;1))=3

3 5

V

V B

V

R

Figure2: Dierentoloringsof H

5

butsameongurationofF

5

anorderingnumbergreaterthani. SeveraloloringsofH

i

mayorrespondto

thesameoloringofF

i

(seegure2). Moreover,theolorsusedbytheverties

V

i nF

i

donotinterferewiththeoloringofthevertieswhihhaveanordering

numbergreaterthani,sinenoedgeexists betweenthem. So,onlythepartial

solutionsorrespondingtodierentoloringsofF

i

havetobestoredinmemory.

Thisway,severalpartialsolutionsonH

i

maybesummarizedbyauniquepartial

solutiononF

i

,alledonguration ofF

i .

The graph oloring problem is solved by evaluating at eah step the on-

gurationsof theboundaryset F

i

. Atstepi, thesubgraph H

i 1

is solved. It

meansthat toeahonguration ofF

i 1

orrespondsavalueoftheminimum

numberofolorsneessarytoolorH

i 1

forthisxedoloringoftheboundary

set verties. Then, partial solutions arebuilt using solutionsof the preedent

step. Thispointwillbedetailedin setion3.4.

3.3 Boundary set ongurations

Aongurationof theboundarysetF

i

is agivenoloringofthevertiesof F

i .

ThisanberepresentedbyapartitionofF

i

,denotedB

1

;:::;B

j

,suhthattwo

verties u;v of F

i

arein thesameblok B

ifand only if theyhavethe same

olor. ThenumberofongurationsofF

i

dependsobviouslyonthenumberof

edgesbetweenthevertiesofF

i

. Theminimumnumberofongurationsis1. If

thevertiesofF

i

formalique,onlyoneongurationispossible: B

1

;:::;B

jFij ,

withexatlyonevertexin eah blok. Themaximalnumberof ongurations

ofF

i

equalsthenumberofpartitionsofasetwithjF

i

jelements. Whennoedge

existsbetweentheboundarysetverties,allthepartitionsaretobeonsidered.

The number of partitions of a set omposed by i elements and j bloks,

A

i;j

j=1 j=2 j=3 j=4

i=1 1[1℄

i=2 1[12℄ 2[1℄[2℄

i=3 1[123℄ 2[13℄[2℄ 5[1℄[2℄[3℄

3[1℄[23℄

4[12℄[3℄

i=4 1[1234℄ 2[134℄[2℄ 9[14℄[2℄[3℄ 15[1℄[2℄[3℄[4℄

3[13℄[24℄ 10[1℄[24℄[3℄

4[14℄[23℄ 11[1℄[2℄[34℄

5[1℄[234℄ 12[13℄[2℄[4℄

6[124℄[3℄ 13[1℄[23℄[4℄

7[12℄[34℄ 14[12℄[3℄[4℄

8[123℄[4℄

writtenA

i;j

,isgivenbythereursiveformulaofStirlingnumbersoftheseond

kind:

A

i;j

=jA

i 1;j +A

i 1;j 1

withA

1;1

=1andA

i;j

=0ifi<j.

ThenumberT(F

i

)ofdierentpartitions oftheboundarysetF

i

equalsthe

sumoftheA

jFij;j

forj from 1tojF

i j.

T(F

i )=

X

j=1tojFij A

jFij;j

Toidentifytheongurationsoftheboundaryset, weassoiatetoeahone

anorderingnumberbetween1andT(F

i

). Thepartitions ofsets withat most

fourelementsandtheirorderingnumberarereportedin table1.

Let C(H

i

;x) be the x th

onguration of F

i

for the subgraph H

i

. Its value,

denotedval(C(H

i

;x)),equalstheminimumnumberofolorsneessarytoolor

H

i

forthisonguration.

Ingure2,twodierentoloringsofH

5

orrespondtothesameonguration

ofF

5

. Only2olorsareneessarytoolorH

5

withtheongurationforwhih

verties2and3haveasameolor,whereas3olorsareneessarywhenverties2

and3havedierentolors. ThevalueoftheongurationC(H

5

;1),represented

bythepartition[45℄,is2,beausewekeeponlythebestvaluation.

The details of theimplementation of the deomposition method are reported

in algorithm 1. Note that H

1

= (f1g;;) and F

1

= f1g. So, there is only

oneongurationof F

1 , C(H

1

;1) =[1℄. Theinsertionof thevertex2 ofG in

C(H

1

;1)anprovideoneortwoongurationsofF

2

(onlyoneongurationif

theverties1and2areneighbors,twoongurationsotherwise).

Atstepi,wedonotexamineallthepossibleongurationsofthestepi 1,

but only those whih have been reatedat preedent step, it meansthose for

whih there is noedge betweentwovertiesof thesame blok. Foreah on-

guration of F

i 1

, we introdue the vertex i in eah bloksuessively. Eah

timetheintrodutionispossiblewithoutbreakingtheoloringrules,theorre-

sponding ongurationof F

i

is generated. Wegenerate alsotheongurations

obtainedby adding to eah ongurationof F

i 1

a new blok ontaining the

vertexi.

For agivensubgraphH

i

,onlytheongurationsthataredierentarerep-

resented. Theirorderingnumberx,inludedbetween1andT(F

i

),isomputed

byan algorithmaordingto their numberof bloks andtheir numberof ele-

ments. WhendierentoloringsofH

i

orrespondtothesameongurationof

F

i

,onlythebestvaluationiskeptin val(C(H

i

;x)).

AtstepN,onlyoneongurationC(H

N

;1)isgeneratedfromongurations

ofstepN 1. Itrepresentsalltheoptimaloloringsolutionsanditsvalueequals

(G).

Exampleofonguration omputing.

AssumethatwearesearhingforaoloringofthegraphGofgure3andthat

weareatstepiwithF

i

=fu;v;w;ig.

Supposethatatstepi 1,wehadF

i 1

=fu;v;wgandthattheongurations

ofF

i 1 were:

-C(H

i 1

;2)=[uw℄[v℄withvalue.

-C(H

i 1

;4)=[uv℄[w℄withvalue.

-C(H

i 1

;5)=[u℄[v℄[w℄ withvalue.

The values of and are at least 2, sine the orresponding ongurations

have2bloks. Remark that these valuesmaybeupper than2, depending on

theongurationsofthepreeedingsteps. Bythesameway,isatleast 3.

WewanttogeneratetheongurationsofF

i

fromtheongurationsofF

i 1 .

- it is impossible to insert i in the rst blok of C(H

i 1

;2), sine u and

i are neighbors. It is possible to insert i in the seond blok of C(H

i 1

;2).

We obtain C(H

i

;3) = [uw℄[vi℄ and val(C(H

i

;3)) = . It is also possible to

Input: agraphG

Output: : thehromatinumberofG

H

1

=(f1g;;)

F

1

=f1g

C(H

1

;1)=[1℄

fori=2toN do

BuildH

i andF

i

foreahongurationC(H

i 1

;x)ofF

i 1 do

forj =1tonumberofbloksofC(H

i 1

;x)do

if idoesnothaveanyneighborintheblokj then

part=C(H

i 1

;x)

insertiin theblokj ofpart

generatetheongurationC(H

i

;y)orrespondingtopart

if C(H

i

;y)alreadyexiststhen

val(C(H

i

;y))=min(val(C(H

i

;y));val(C(H

i 1

;x)))

else

val(C(H

i

;y))=val(C(H

i 1

;x))

endif

endif

endfor

part=C(H

i 1

;x)

addto part anewblokontainingi

val(part)=max(val(C(H

i 1

;x)),numberofbloksofpart)

generatetheongurationC(H

i

;y)orrespondingtopart

if C(H

i

;y)alreadyexiststhen

val(C(H

i

;y))=min(val(C(H

i

;y));val(part))

else

val(C(H

i

;y))=val(part)

endif

endfor

endfor

=val(C(H

N

;1))