Pre-processings and Linear-Decomposition Algorithm to Solve the k-Colorability Problem

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Pre-processings and Linear-Decomposition Algorithm to Solve the k-Colorability Problem

Corinne Lucet, Florence Mendes, Aziz Moukrim

To cite this version:

Corinne Lucet, Florence Mendes, Aziz Moukrim. Pre-processings and Linear-Decomposition Algo- rithm to Solve the k-Colorability Problem. International Workshop on Experimental and Efficient Algorithms WEA’2004, May 2004, Brazil. pp.315-325. �hal-00783886�

Algorithm to Solve the k-Colorability Problem

?

C.Luet 1

,F.Mendes 1

,A. Moukrim 2

1

LaRIAEA2083,5rueduMoulin Neuf80000Amiens,Frane

2

HeuDiaSyCUMRCNRS6599UTC,BP20529 60205Compiegne,Frane

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

Aziz.Moukrimhds.ut.fr

Abstrat. Weare interestedinthe grapholoring problem.We stud-

iedtheeetiveness ofsome pre-proessingsthat are speito the k-

olorabilityproblemandthatpromisetoreduethesizeorthediÆulty

of the instanes. We propose to apply onthe redued graph an exat

methodbasedonalinear-deompositionofthegraph.Wepresentsome

experimentsperformed onliterature instanes, among whihDIMACS

libraryinstanes.

1 Introdution

The Graph Coloring Problem onstitutes a entral problem in a lot of appli-

ations suh asshool timetabling, sheduling, or frequeny assignment [5,6℄.

This problembelongsto thelassofNP-hardproblems[10℄.Variousheuristis

approahes have been proposed to solve it (see for instane [2,8,9,11,13,17,

19,21℄).EÆientexatmethodsarelessnumerous:impliitenumerationstrate-

gies [14,20,22℄, olumn generation and linear programming [18℄, branh-and-

bound [3℄,branh-and-ut[7℄, withoutforgettingthe well-knownexatversion

ofBrelaz'sDSATUR[2℄.

A oloring of a graphG =(V;E) is an assignmentof a olor(x) to eah

vertex suh that (x) 6= (y) for all edges (x;y) 2 E. If the number of olors

used is k,the oloringof Gisalled ak-oloring. Theminimumvalue ofk for

whihak-oloringispossibleisalledthehromatinumberofGandisdenoted

(G). Thegraph oloring problem onsistsin ndingthe hromati numberof

agraph.Ourapproahtosolvethis problemistosolvefordierentvaluesofk

thek-olorability problem: \doesthereexist ak-oloringofG?".

Weproposetoexperimenttheeetivenessofsomepre-proessingsthatare

diretlyrelatedtothek-olorabilityproblem.Theaimoftheseproessingsisto

redue thesize of the graphby deleting vertiesandto onstrain it by adding

edges. Then we apply a linear-deomposition algorithm on the reduedgraph

in order to solvethe graph oloringproblem. This method is strongly related

tonotionsoftree-deomposition andpath-deomposition,wellstudiedbyBod-

laender [1℄. Linear-deomposition has been implemented eÆiently by Carlier,

?

main advantage that the exponential fator of its omplexity depends on the

linearwidthofthegraphbut notonitssize.

Our paper is organized as follows. We present in Set. 2 somepre-proes-

singsrelatedtothek-olorabilityproblemandtesttheireetivenessonvarious

benhmarkinstanes.InSet.3,wedesribeourlinear-deompositionalgorithm.

We report the resultsof our experiments in Set. 4. Finally, we onlude and

disussabouttheperspetivesofthiswork.

2 Pre-proessings

Inthissetion,wepresentseveralpre-proessingstoreduethediÆultyofak-

olorabilityproblem.Thesepre-proessingsareiterateduntilthegraphremains

unhangedorthewholegraphisredued.

2.1 Denitions

An undireted graph Gis apair(V;E)madeupof avertexset V and anedge

set E V V. Let N = jVj and M = jEj. A graph G is onneted if for

all vertiesw;v 2 V(w 6=v),there exists a pathfrom w to v. Without lossof

generality,thegraphswewillonsiderinthefollowingofthispaperwillbeonly

undiretedandonnetedgraphs.GivenagraphG=(V;E)andavertexx2V,

let#(x)=fy2V=(x;y)2Eg.#(x)representstheneighborhoodofx inG.The

subgraphofG=(V;E)induedbyI V,isthegraphG(I)=(I;E

I

)suhthat

E

I

=E\(II). A lique of G= (V;E)is a subset C V suh that every

twovertiesin C arejoined byan edgein E. LetE =(V V)nE bethe set

madeupof allpairsofvertiesthat arenotneighbors in G=(V;E).Letd be

thedegreeofG,i.e.themaximalvertexdegreeamongallvertiesofG.

2.2 Redution 1

A vertex redution using the following property of the neighborhood of the

vertiesanbeappliedtotherepresentativegraphbeforeanyotheromputation

with time omplexityO(jEjd), upperbounded byO(N 3

). Given agraphG,

for eah pair of verties x;y 2 V suh that (x;y) 2= E, if #(y) #(x) then

y and its adjaent edges an be erased from the graph. Indeed, suppose that

k 1olors areneeded toolor theneighbors of x.Thevertexx antakethe

k th

olor.Vertiesx andy are notneighbors.Moreover,theneighborsof y are

already oloredwith atmost k 1olors. So,if Gnfygis k-olorablethen G

isk-olorableaswellandweandeletey fromthegraph.Thisprinipleanbe

applied reursivelyaslongasvertiesareremovedfrom thegraph.

2.3 Redution 2

Supposethatwearesearhingforak-oloringofagraphG=(V;E).Thenwe

has k 1 neighbors. In the worst ase, those neighbors must have dierent

olors. Then the vertex x an take the k th

olor. It does not interfere in the

oloring of the remaining verties beause all its neighbors havealready been

olored.Therefore weanonsider fromthe beginningthat it will takeaolor

unused by its neighbors and delete it from the graphbefore the oloring.The

time omplexity ofthis redutionis O(N). We applythis priniplereursively

by examiningthe remainingvertiesuntil havingtotally reduedthegraph or

beingenabletodeleteanyothervertex.

2.4 VertexFusion

Supposethat wearesearhingforak-oloringofG=(V;E)and thatalique

C of size k has been previously determined. For eah ouple of non-adjaent

verties x;y 2 V suh that x 2= C and y 2 C, if x is adjaent to all verties

of Cny then x and y an be merged by the following way: eah neighbor of

x beomes aneighborof y, then x and its adjaentedges are erased from the

graph. Indeed,sine weare searhing for a k-oloring,x and y must havethe

sameolor.Then8z2#(x)(y)6=(z)andtheedge(y;z)anbeadded toG.

Then #(x)#(y)andx an beerasedfromthe graph(f Set.2.2). Thetime

omplexityofthispre-proessingisO(Nk).

2.5 Edge Addition

Supposethatwearesearhingforak-oloringofG=(V;E)andthataliqueC

ofsizekhasbeenpreviouslydetermined.Foreahoupleofnon-adjaentverties

x;y2V,if8z2C wehave(x;z)2Eor(y;z)2E,thentheedge(x;y)anbe

addedtothegraph.Neessarly,xmusttakeaolorfromtheolorsofCn#(x).

Sine #(y) Cn#(x), (x) 6=(y). This onstraint anbe representedby an

edgebetweenxandy.Thetimeomplexityofthispre-proessingisO(jEjk),

upperboundedbyO(N 2

k).

Algorithm1Pre-proessings

Input: agraphGandanintegerk

Output: agraphG 0

k-olorableifandonlyifGisk-olorable

repeat

redution1

redution2

if 9atleast1liqueofsizekthen

applyvertexfusionandedgeadditiononG

endif

untilthereisnomorehangeinG

0

Ouralgorithms havebeenimplementedon aPCAMD AthlonXp 2000+in C

language.Themethod used isasfollows.Tostart with,weapplyon theentry

graphGafastliquesearhalgorithm:aslongasthegraphisnottriangulated,

weremoveavertexofsmallestdegree,andthenweolortheremainingtriangu-

latedgraphbydeterminingaperfeteliminationorder[12℄onthevertiesofG.

Thesizeoftheliqueprovidedbythisalgorithm,denotedLB,onstitutesalower

bound ofthehromati numberof G.ThenweapplyonGthepre-proessings

desribedinAlgorithm1,supposingthatwearesearhingforak-oloringofthe

graph with k = LB. We performedtests onbenhmark instanes used at the

omputationalsymposiumCOLOR02,inludingwell-knownDIMACSinstanes

(see desription of the instanes at http://mat.gsia.mu.edu/COLOR02). Re-

sults are reported in Table 1. For eah graph, we indiate the initial number

of verties N and the numberof edges M. The olumn LB ontains the size

of the maximal lique found. The perentage of verties deleted by the pre-

proessings isreported in olumn Del. Thenumberof remainingvertiesafter

thepre-proessingstepisreportedin olumnnewN.Remarkthat someofthe

instanes are totally redued by the pre-proessings when k = LB, and that

someofthemarenotreduedatall.

Table1:Pre-proessingsresults

Graph N M LBnewN Del Graph N M LBnewN Del

1-FullIns3 30 100 3 15 50% 1-FullIns4 93 593 3 35 62%

1-FullIns5 282 3247 3 75 73% 2-FullIns3 52 201 4 9 81%

2-FullIns4 212 1621 4 41 81% 2-FullIns5 85212201 4 89 90%

3-FullIns3 80 346 5 11 86% 3-FullIns4 405 3524 2 51 87%

3-FullIns5 203033751 2 107 95% 4-FullIns3 114 541 6 13 89%

4-FullIns4 690 6650 2 58 92% 5-FullIns3 154 792 7 15 90%

5-FullIns4 108511395 2 65 94% fpsol2.i.1 49611654 65 228 54%

fpsol2.i.2 451 8691 30 175 61% fpsol2.i.3 425 8688 30 149 65%

inithx.i.1 864 18707 54 443 49% inithx.i.2 64513979 31 215 67%

inithx.i.3 621 13969 31 190 69% mulsol.i.1 197 3925 49 60 70%

mulsol.i.2 188 3885 31 88 53% mulsol.i.3 184 3916 31 83 55%

mulsol.i.4 185 3946 31 85 54% mulsol.i.5 186 3973 31 84 55%

shool1 385 19095 14 360 6% shool1nsh 35214612 14 331 6%

3-Inser3 56 110 2 56 0% 4-Inser3 79 156 2 79 0%

le450 25a 450 8260 20 297 34% le450 25b 450 8263 25 294 35%

anna 138 493 11 0 100% david 87 812 11 0 100%

homer 561 1629 13 0 100% jean 80 508 10 0 100%

mug100-1 100 166 3 100 0% mug100-25 100 166 3 100 0%

mug88-1 88 146 3 88 0% mug88-25 88 146 3 88 0%

miles250 128 387 7 34 73% miles500 128 2340 20 0 100%

miles750 128 4226 31 0 100% miles1000 128 6432 42 0 100%

miles1500 128 10396 73 0 100% DSJR500 1 500 3555 12 28 94%

zeroin.i.1 211 4100 49 86 59% zeroin.i.2 211 3541 30 55 74%

Problem

In this setion, we propose a method whih uses linear-deomposition mixed

withDsaturheuristiinordertosolvethek-olorabilityproblem.

3.1 Denitions

WewillonsideragraphG=(V;E).LetN =jVjandM=jEj.Avertexlinear

orderingofGisabijetionN :V !f1;:::;Ng.Formorelarity,wedenoteithe

vertexN 1

(i).LetV

i

besubsetofV madeofthevertiesnumberedfrom1toi.

LetH

i

=(V

i

;E

i

)bethesubgraphofGinduedbyV

i .LetF

i

=fj2V=9(j;l)2

E j i <lg8i2f1;:::;jVjg. F

i

is theboundaryset of H

i .LetH

0

i

=(V 0

i

;E 0

i )

bethesubgraphofGsuhthatV 0

i

=(V nV

i )[F

i andE

0

i

=E\(V 0

i V

0

i ).The

boundarysetF

i

orrespondstothesetofvertiesjoiningH

i toH

0

i

(seeFig.1).

3 9

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

Fig.1.AsubgraphH10ofGanditsboundarysetF10=f7;8;10g

Thelinearwidth of avertex linearorderingN is F

max

(N)=max

i2V (jF

i j).

insertion of theverties, using avertex linearordering previously determined.

Thiswillbedevelopedin thefollowingsetion.

3.2 Linear-DeompositionAlgorithm

The details of the implementation of the linear-deomposition method are re-

ported in Algorithm 2. The verties of G are numbered aording to alinear

ordering N : V ! f1;:::;Ng. Then, during the oloring, we will onsider N

subgraphs H

1

;:::;H

N

and theN orresponding boundary sets F

1

;:::;F

N , as

dened inSet.3.1.

Algorithm2k-olorability

Input: agraphGandanintegerk

Output: R esul t:TrueifandonlyifGisk-oloriable

H1=(f1g;;)

F1=f1g

C(H1;1)=[1℄

i=2

R esul t=True

whileiN andR esul tdo

R esul t=Fal se

BuildH

i andF

i

foreahongurationC(Hi 1;x)ofFi

1 do

forj=1tonumberofbloksofC(Hi 1;x)do

if idoesnothaveanyneighborintheblokjthen

R esul t=True

part=C(H

i 1

;x)

insertiintheblokj ofpart

generatetheongurationC(H

i

;y)orrespondingtopart

val (C(Hi;y))=min(val (C(Hi;y));val (C(Hi 1;x)))

endif

endfor

if numberofbloksofC(Hi 1;x)<kthen

R esul t=True

part=C(Hi 1;x)

addtopartanewblokontainingi

val (part)=max(val (C(Hi 1;x)),numberofbloksofpart)

generatetheongurationC(H

i

;y)orrespondingtopart

val (C(Hi;y))=min(val (C(Hi;y));val (part))

endif

endfor

i=i+1

endwhile

The omplexity of the linear-deomposition is exponential with respet to

F (N) ,soitisneessarytomakeagoodhoiewhennumberingtheverties

to obtain the smallest linearwidth is a NP-omplete problem [1℄. After some

experimentson various heuristisof vertex numbering, wehooseto beginthe

numbering from the biggest lique provided by our lique searh heuristi (f

Set.2.6).Thenweorderthevertiesbydereasingnumberofalreadynumbered

neighbors.

Startingfromavertexlinearordering,webuildatrstiterationasubgraph

H

1

whih ontainsonlythevertex1,thenat eah stepthenextvertexandits

orresponding edges are added, until H

N

. Toeah subgraphH

i

orrespondsa

boundarysetF

i

ontainingthevertiesofH

i

whihhaveatleastoneneighborin

H 0

i

.TheboundarysetF

i

isbuiltfromF

i 1

byaddingthevertexiandremoving

the verties whose neighbors have all been numbered with at most i. Several

oloringsofH

i

mayorrespondtothesameoloringofF

i

.Moreover,theolors

usedbythevertiesV

i nF

i

donotinterferewiththeoloringofthevertieswhih

haveanorderingnumbergreaterthani,sinenoedgeexistsbetweenthem.So,

only thepartial solutionsorrespondingto dierentoloringsof F

i

have to be

storedinmemory.Thisway,severalpartialsolutionsonH

i

maybesummarized

byauniquepartialsolutiononF

i

,alled onguration ofF

i .

Aongurationoftheboundaryset F

i

isagivenoloringofthevertiesof

F

i

.Thisanberepresentedbyapartition ofF

i

,denotedB

1

;:::;B

j

, suh that

two verties u;v of F

i

are in the same blok B

if they have the same olor.

Thenumberof ongurationsof F

i

depends obviouslyonthenumberof edges

betweenthe vertiesof F

i

.The minimumnumberof ongurationsis1. Ifthe

verties of F

i

form a lique, only one onguration is possible: B

1

;:::;B

jF

i j

,

with exatly onevertexin eah blok. The maximal numberof ongurations

ofF

i

equalsthenumberofpossiblepartitionsofasetwithjF

i

jelements.When

no edge exists between the boundary set verties, all the partitions are to be

onsidered. Their number T(F

i

) grows exponentially aording to the size of

F

i

.Theirorderingnumberx,inludedbetween1andT(F

i

),isomputedbyan

algorithmaordingtotheirnumberofbloksandtheirnumberofelements.This

algorithm uses the reursive prinipleof Stirling numbers of the seond kind.

The partitions of sets with at most four elements and their ordering number

are reported in Table 2. Let C(H

i

;x) be the x th

onguration of F

i

for the

subgraph H

i

. Its value, denoted val(C(H

i

;x)) equalsthe minimum number of

olorsneessarytoolorH

i

forthisonguration.

Atstepi,fortunatelywedonotexamineallthepossibleongurationsofthe

stepi 1,but onlythosewhih havebeenreatedat preedentstep,itmeans

those for whih there is no edge between two verties of the same blok. For

eahongurationofF

i 1

,weintroduethevertexi ineahbloksuessively.

Eah timethe introdution ispossiblewithoutbreaking theoloringrules,the

orrespondingongurationofF

i

isgenerated.Moreover,foreahonguration

ofF

i 1

withvaluestritlylowerthank 1,wegeneratealsotheonguration

obtainedbyaddinganewblokontainingthevertexi.

Inordertoimprovethelinear-deomposition,weapplytheDsaturheuristi

evenly onthe remaininggraphH 0

, for dierentongurationsof F

i

.If Dsatur

j=1 j=2 j=3 j=4 T(i)

i=1 1[1℄ T(1)=1

i=2 1[12℄ 2[1℄[2℄ T(2)=2

i=3 1[123℄ 2[13℄[2℄ 5[1℄[2℄[3℄

3[1℄[23℄

4[12℄[3℄ T(3)=5

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℄ T(4)=15

ndsak-oloringthen theproess endsand theresult ofthek-oloringis yes.

Otherwisethelinear-deompositionontinuesuntilaongurationisgenerated

at step N, in this ase the graph is k-olorable, or no onguration an be

generatedfromthepreedentstep,inthisasethegraphisnotk-olorable.The

omplexityofthelinear-deompositionalgorithm,upperboundedbyN2 Fmax

,

isexponentialaordingtothelinearwidthofthegraph,butlinearaordingto

itsnumberofverties.

3.3 Exampleof Conguration Computing

Assume that we are searhing for a 3-oloring of the graphG of Fig. 2. Sup-

pose thatat step i 1we hadF

i 1

=fu;vg.Theongurationsof F

i 1 were

C(H

i 1

;1)=[uv℄ofvalueandC(H

i 1

;2)=[u℄[v℄ ofvalue.Thevalueof

is2or3,sinetheorrespondingongurationhas2bloksandk=3.

u

v

i

H

i 1

G

Fig.2.ConstrutionofH =(V [fig;E [f(u;i)g)