Publicações do PESC Two Novel Evolutionary Formulations of the Graph Coloring Problem

of the Graph Coloring Problem Valmir C.Barbosa 1 CarlosA. G.Assis 2 Josina O.do Nascimento 3 Abstract

We introduce two novel evolutionary formulations of the problem of coloring the nodes of a graph. The

rstformulationisbasedontherelationshipthatexistsbetweenagraph'schromaticnumberanditsacyclic

orientations. It views such orientations asindividuals and evolves them with the aid of evolutionary

op-erators that areveryheavily basedonthe structure ofthe graphand itsacyclicorientations. The second

formulation, unliketherst one,doesnottackleonegraphat atime, but ratheraims at evolvinga

\pro-gram"tocolorallgraphsbelongingtoaclasswhosemembersallhavethesamenumberofnodesandother

commonattributes. Theheuristicsthatresultfromtheseformulationshavebeentestedonsomeofthe

Sec-ond DIMACS Implementation Challengebenchmark graphs,andhave beenfound to becompetitivewhen

comparedtotheseveralotherheuristicsthathavealsobeentestedonthosegraphs.

Keywords: Graphcoloring,evolutionaryalgorithms,geneticalgorithms,geneticprogramming.

1. Introduction

WeconsideragraphG=(N;E),whereN isthegraph'ssetofnodesandEitssetofedges,eachedgebeing

anunorderedpairofnodes,calledneighbors ofeachother. GraphGissaidtobeanundirected graph,since

theunorderednature ofitsedgesassignsno\directionality"tothem. Weletn=jNjandm=jEj.

Acoloring ofGisanassignmentofcolors (positiveintegers)tothenodesinsuchawaythateverynode

is assigned exactly onecolor and, furthermore, no twoneighbors are assigned the samecolor. The graph

coloring problem is theproblem of providing G with acoloringthat employs theleast possible numberof

colors. Thisnumberiscalledthechromaticnumber ofG,denotedby(G).

Thereasonswhythegraphcoloringproblemisimportantaretwofold. First,thereareseveral areasof

practicalinterestinwhichtheabilitytocoloranundirectedgraphwithassmallanumberofcolorsaspossible

hasdirectin uenceonhoweÆcientlyacertaintargetproblemcanbesolved. Suchareasincludetimetabling

andscheduling[12,29],frequencyassignmentforuseoftheelectromagneticspectrum[19],registerallocation

1

UniversidadeFederaldoRiodeJaneiro,ProgramadeEngenhariadeSistemaseComputac~ao,COPPE,

CaixaPostal68511,21945-970RiodeJaneiro-RJ,Brazil. E-mail: [email protected].

2

PontifciaUniversidadeCatolicadoRiodeJaneiro,DepartamentodeInformatica,RuaMarqu^esdeS~ao

Vicente,225,22453-900Riode Janeiro- RJ,Brazil, and Universidade Estaciode Sa, AvenidaAutomovel

Clube,126,20765-000RiodeJaneiro-RJ,Brazil. E-mail: [email protected].

3

ObservatorioNacional,RuaGeneralJoseCristino,77,20921-400RiodeJaneiro-RJ,Brazil,and

nite-elementmeshes[35].

Theother reasonisthat thegraphcoloringproblem hasbeenshownto becomputationallyhard ata

varietyoflevels: notonlyisitsdecision-problemvariantNP-complete[20,27],butalsoitisNP-hardtosolve

itapproximatelywithin n 1=7 

oftheoptimumfor any>0[1,2,6]. Whatthis meansis that, giventhe

currentexpectationofhowlikelyitisforanalgorithmto befoundtosolveanNP-hardproblemeÆciently,

coloring the nodes of G with at most n 1=7 

(G) colors is an intractable problem for any  > 0. There

isasense,then, in which graphcoloringstandsamongthe hardestof thehardcombinatorial optimization

problems.

These tworeasonsare importantenoughtojustify thequestfor heuristicstosolvethe graphcoloring

problem,andtoplaceittogetherwiththefavoriteproblemsonwhichnewmeta-heuristicsaretested. Many

such approaches have been proposed (e.g., [25] and the references therein), but we single oneout which,

althoughitmissesthegraph'schromaticnumberbyratherawidemargininseveralcases,hasanimportant

inspirationalroleinpartofthispaper. Thisheuristicisbasedonthenodes'degrees (numbersofneighbors)

and onthe knownpropertythat (G)
+1, where
is the maximumdegree overall nodes [7]. The

heuristic visits nodes in nonincreasing order of the so-called saturation degree (number of colors already

used to color neighbors) and assigns thecurrentnodethe smallestcolor that is not alreadytaken by one

ofitsneighbors [8]. Symmetryisbrokenbyusing anonincreasingorder ofdegreesto which onlyneighbors

that havenotyetbeencoloredcontribute,andafter thatrandomly. ThisheuristicisknownastheDSatur

heuristic.

Despitetheproblem'simportanceandthemanyapproachesthathavebeentriedtotackleit,itwasnot

untilrelativelyrecentlythataconcentratedeortwasundertakentotestseveralheuristicsonacommonset

ofgraphsofvarioussizesandcharacteristics. SuchaneortwaspartoftheSecondDIMACSImplementation

Challenge[26],whichremainsto datethemostcomprehensiveprojectthatweknowoftohaveaddresseda

systematicexperimentalevaluationofheuristicsforthegraphcoloringproblem.

Comprehensivethoughthis eortwas,onlyoneoftheapproachestomakeitto thenalmeeting was

of an evolutionary nature [16]. This approach involved the combination of evolutionary and other search

techniques, and was based on previous work by the same authors [15]. In fact, to our knowledge the

situation remains that no purely evolutionary approach seems to havehad success on the graph coloring

problem,althoughtherehavebeenevolutionaryapproachesforrestrictedversionsoftheproblem(e.g.,[14])

andalsootherhybridapproacheswithevolutionaryingredients(e.g.,[18]andthereferencestherein). Inour

view, the central diÆculty is the diÆculty of formulating the problem adequatelyso that individuals and

the appropriate evolutionary operators canbe properly identied. Suppose, for example, that we choose

to approachthe formulationin the obvious, straightforwardmanner, and say that anindividual is simply

aparticularcoloring. That is sound enough,but howdoesone goabout, say, generating anewindividual

from twoparentcolorings? Coloring somenodes accordingto one parentand theothers accordingto the

otherparentmightwork,butnotwithoutsomeadditionalruletohandle thecasesinwhichthesamecolor

ends upassignedto neighbors,orelse allowingsuch infeasible colorassignmentswhilesomehowpenalizing

themthroughappropriatelylowtnessvalues.

Asweperceiveit,theproblemwiththisnaverstapproachandotherscloselyrelatedtoitisthesame

that preventsmassiveparallelismfrom workingsatisfactorilyon thegraphcoloringproblem under certain

other meta-heuristics [5]: the problem speaks of aglobal property of graphs(how many colors overall?),

correspondingsubsetofEforedgeset). Inparticular,ifG

1 andG

2

aredistinctsubgraphsofG,thenitdoes

notfollowthat(G)=(G

1

)+(G

2 ).

Notwithstanding such diÆculties, graph coloringis a research area in which much of the underlying

structure hasalreadybeenuncoveredandconsiderable heuristicexperiencehas beenaccumulated. Inthis

paper,wetakeadvantageofsomekeyaspectsofthis structureandknowledgetointroducetwonew

evolu-tionaryformulationsofthegraphcoloringproblem. Therstformulationbelongsintheclassoftechniques

wenormally thinkof asgenetic algorithms[24, 33], whilethesecond onecanbethoughtof as aninstance

ofgeneticprogramming[3,28].

Ourrstapproach,describedinSection2,isbasedonaviewofthecoloringsofGthatrelatesthemto

theacyclicorientations ofG. An acyclicorientationofGisanyofthepossiblewaysofassigningdirections

tothe(undirected)edgesofGinsuchawaythatnodirectedcycle isformed(thatis,insuchawaythat,by

traversingedgesonlyaccordingtotheassigneddirectionsitisimpossibletoreachthesamenodemorethan

once). Ingeneral,anorientationofGcanberegardedasafunction!:E !N suchthat,fore=(i;j)2E,

!(e)=j ifandonlyifedgeeis directedfromnodei tonodej.

Let adirected path in G accordingto some acyclicorientation bea sequenceof nodes, each of which

is a neighbor of its predecessor, if it has one, such that the last node can be reached from the rst by

traversing the edges that connect consecutive nodes to each other along the directions assigned to them

by the orientation. The essence of our rst approach is that, to any coloring that uses c colors overall

there corresponds anacyclicorientationwhose longestdirected path hasno morethanc nodes; and that,

conversely,to anyacyclicorientationwhoselongest directedpath haslnodesthere corresponds acoloring

bynomorethanlcolors. Ifwelet(G) denotetheset ofallacyclicorientationsof G,thenthis givesrise

tothefollowingrelation between(G)and(G). For!2(G), weletP(!)denotetheset ofalldirected

pathsinGaccordingto !. Forp2P(!),letl(p) bethenumberofnodesinp. Then,from[13],wehave

(G)= min

!2(G) max

p2P(!)

l(p): (1)

Our rst approach regards (G) as the search space out of which populations are formed, that is, each

acyclic orientationof G is an individual. The ttest possibleindividual is the onewhose longest directed

pathisshortestamongallpossibleindividuals.

ThisrstapproachworksonxedG,thatis,everynewgraphthatcomesalongtobecoloredtriggersthe

evolutionofacyclicorientationsinordertondoneofhightness. Thesecondapproach,whichwedescribe

inSection3,isbycontrasttargetedatanentireclassGofundirectedgraphssharingcertaincharacteristics,

asfor examplethe valueof n. It is inspired bythe DSatur heuristicdiscussed above, in the sense that it

approachesthecoloringofG'snodesasthesearchforadegree-basedorderingofthosenodesthatwillyield

agoodcoloringifnodesarecoloredasdictatedbythatordering.

Morespecically,supposeforthemomentthatthenumberofnodesisthesolecharacteristicsharedby

themembersofG,andletK(n)standforthesetofalln!permutationsofthesequenceh1;:::;ni. Amember

=hk

1 ;:::;k

n

iof K(n)canbeinterpretedasthefollowingsequenceofinstructionsto assigncolorstothe

nodesofanygraphG2G: rstassigncolor1tothenodehavingthek

1

thlargestdegree,thenassigntothe

nodehavingthek

2

thlargestdegreethesmallestcolorthatdoesnotcon ictwiththecolorspossiblyalready

assignedtoitsneighbors,andsoon. Ifwefornowdisregardtheexistenceofnodeshavingthesamedegree,

thenK(n)isthesearchspacethat containsthepopulationsofoursecondevolutionaryapproach. Fitnessis

inthiscaseanaveragetakenoverapre-selectedsubclassofG. Anindividualtterthananotherwillrequire

Section 4, abrief discussionon howthe twostrategies' evolutionaryoperatorsrelate to their statespaces

is presented. Section 5 contains a description of the graphs that we use in our experiments, which are

drawnfromthesetoftestgraphsemployedintheDIMACSimplementationchallenge. Experimentalresults

are given in Section 6. They indicate that our two new evolutionary formulations yield algorithms that

are competitive when compared to several others, including those developed in response to the DIMACS

challenge. Section7containsconcludingremarks.

2. The rst formulation

Ourrstformulationis basedontheinterplaybetweenacyclicorientationsofGandcoloringsofitsnodes.

Ifacoloringexiststhat usesc colorsoverall,thenconsidertheorientationobtainedbydirectingeveryedge

fromthenodewiththehighestcolortotheonewiththelowest. Theresultingorientationisclearlyacyclic,

and furthermoreno directed path exists havingmorethan c nodes. Inthe samevein, consider anacyclic

orientationwhoselongestdirectedpathhaslnodes,andconsiderthenodesthataresinks accordingtothis

orientation (a sinkis anode whose incidentedges are alldirected inwardbythe orientation). Nowassign

colorsto nodesasfollows. Firstassign color1to allsinks. Thenremoveallsinksfrom G(togetherwithall

edgesincidenttothem) andassigncolor2to allthenewsinks thatarethusformed. Thisremovalofsinks

canbe repeated exactly l times, and each time it is repeated the resulting sinks are assigned their color,

whichconceivablycanbe someof thecolorsusedpreviously. What resultsisthenacoloringthat employs

nomorethanlcolors.

An important consequence of this interplay is (1), which relates the chromatic number of G to its

acyclicorientations. Ourrst formulationregardsacyclic orientationsasindividuals and aimsat evolving

anacyclicorientationwhose longestdirected pathisasshortas possible. Oncethisindividual isidentied,

thecorrespondingcoloringofthenodesofGcanbefoundasindicatedpreviously.

Beforewegivethedetailsofthisformulation,wepausemomentarilyto commentonthecardinalityof

(G),thesetofallacyclicorientationsofG. Whileforsomegraphssuchcardinalityisknownasafunction

ofnormthatcanbecomputedeasily(2 n 1

fortrees,n!forcompletegraphs,2 n

2forrings),thegeneral

caserequirestheevaluationoftheso-calledchromaticpolynomial ofG. Forc>0,thispolynomial,denoted

byC

G

(c),yieldsthenumberofdistinctwaysinwhichthenodesofGcanbecoloredbyatotalofatmostc

colors[7]. Clearly,thesmallestsuchcforwhichC

G

(c)>0isequalto(G).

Oncethechromaticpolynomialof Ghasbeenintroduced,thecardinalityof(G)canbeprovento be

givenby   (G)   =( 1) n C G ( 1);

asdemonstratedin[37]. Wethinkthisisaremarkableproperty,andfeltitshouldbepresentedeventhough

it bears no further relationship to thetopic of thepaperthan clarifyinghowto assess   (G)   , at least in

principle, in the generalcase, and perhapsproviding additional evidence that coloringsof G's nodes and

acyclicorientationsofGareindeed deeplyrelatedto oneanother.

Theremainderof thissectioncontainsadetailed descriptionofthekeyelementsofourrstapproach.

These are the assessment of an acyclic orientation's tness, and the functioning of the two evolutionary

operators we employ, namely crossoverand mutation. We use thefollowingadditional notation. For ! 2

(G), weletS +

(!)be thesetof sinksin Gaccordingto !. Likewise, S (!)denotestheset ofsources in

Gaccordingto! (these arenodeswhoseincident edgesarealldirected outwardby!). Fori2N, N

i (!)

1 f 1 (!)=n max p2P(!) l(p); (2)

wherewerecallthatP(!)isthesetofalldirectedpathsinGaccordingto!andl(p)isthenumberofnodes

in directed path p. It followsfrom (2) that an individual has higher tnessthan another if its lengthiest

directed pathis shorterthantheother's. Byourpreceding discussion, anindividual! yieldsacoloringof

G'snodesbynomorethann f

1

(!)colors.

Clearly,anequivalentreformulationof f

1 (!)is f 1 (!)=n max i2S (!) l i ; (3) wherel i

isthenumberofnodesonthelongestdirectedpathinGaccordingto! thatstartsatnodei. The

expressionforf

1

(!)in (3)ismoreusefulbecausethevalueofl

i

foralli2N canbecomputedeÆcientlyby

straightforwarddepth-rstsearch[10],asinproceduredfs(i;!),givennext.

For alli2N, letreached

i

beaBooleanvariable usedto indicatewhether node ihasbeenreachedby

thedepth-rstsearch(itissetto falseinitially). Wethenhave:

proceduredfs(i;!) ifreached i =false reached i :=true ifi2S + (!) l i :=1 else l i :=1+max j2N i (!) dfs(j;!) Returnl i

Thisprocedureallowsustorewritef

1 (!)yetagainas f 1 (!)=n max i2S (!) dfs(i;!); (4)

whichindicatesdirectlyhowthetnessofanindividualistobeevaluated inpractice.

Crossover. Thecrossoveroftwoparentindividuals!

1 ;!

2

2(G)toyieldthetwoospring! 0 1 ;! 0 2 2(G)

is better understood if we adopt a linearrepresentation for the acyclicorientationsof G. For! 2 (G),

saythat anodej isreachable fromnode iaccordingto ! ifadirected pathexistsfrom itoj. Nowletthe

sequenceofnodes L(!)=hi 1 ;:::;i n i

besuchthat,for1x;yn,x<y ifandonlyifeitheri

y

isreachablefrom i

x

orneithernodeisreachable

from the other but i

x <i

y

. Note that here, asin sectionsto come, we assumefor simplicity that the set

of nodesN canbe regardedastheset of naturalnumbersf1;:::;ng,sothat the comparison\i

x <i

y "is

meaningful.

Ifweregard! as apartialorderof thenodesofG,then clearlyL(!)istheuniquelinearextensionof

that partialorder withthefollowingcharacteristic. Fori

x

2N, everyothernode i

y

suchthat x<y (that

is, to the right of i

x

in L(!)) is either reachablefrom i

x

x y x y

betweenthetwonodes,butsomesymmetry-breakingruleisnecessarytoconstructthelinearrepresentation.

InSection6,wecommentbrie yonthepossibleeectsofusingrandomnesstobreaksymmetry.

Wedescribethecrossoverof!

1 and!

2

in termsoftheirlinearrepresentations,namely

L(! 1 )=hi 1 ;:::;i n i and L(! 2 )=hj 1 ;:::;j n i:

Ifz suchthat 1z<nisthecrossoverpointandwelet

L(! 0 1 )=hi 0 1 ;:::;i 0 n i and L(! 0 2 )=hj 0 1 ;:::;j 0 n i; then hi 0 1 ;:::;i 0 z i=hi 1 ;:::;i z iand hi 0 z+1 ;:::;i 0 n i isthesubsequenceof hj 1 ;:::;j n

icomprisingall thenodes

thatarenotalreadyinhi

1 ;:::;i z i. TheconstructionofL(! 0 2 )isentirelyanalogous: hj 0 1 ;:::;j 0 z i=hj 1 ;:::;j z i and hj 0 z+1 ;:::;j 0 n i is the subsequence of hi 1 ;:::;i n

i comprising all the nodes that are not already in

hj

1 ;:::;j

z i.

In orderto givean interpretation ofthis crossoveroperation that is meaningfulin thecontextof the

acyclicorientationsofG,wemustrstrecognizethat thelinearrepresentationweadoptedisunambiguous.

Infact,anysequenceLcomprisingallnodesfromN canbeseentogiverisetoauniqueacyclicorientation,

as follows. For x < y, if i

x and i

y

are neighbors in G, then direct the edge between them from i

x to

i

y

. Any directed cycle in the resultingorientationwould imply there exivity of <, which doesnot hold.

Consequently,wearejustiedinhavingwrittenL(! 0

1

)andL(! 0

2

),becausetheuniqueorientationsthatresult

fromthosesequencesareindeed acyclic.

Notonlythis,butwecannowseeclearlywhatisinheritedby! 0 1 and! 0 2 from! 1 and! 2 . Orientation! 0 1 inheritsfrom! 1

thedirectionsassignedtoalledgesjoining nodesinthesetfi 0

1 ;:::;i

0

z

gtoanyothernodes,

andfrom!

2

thoseassignedtotheedgeswhoseendnodesalllieinfi 0 z+1 ;:::;i 0 n g. Asfor! 0 2 ,itinheritsfrom ! 2

thedirectionsassignedtoalledgesjoiningnodesin thesetfj 0

1 ;:::;j

0

z

gtoanyothernodes,andfrom!

1

thoseassignedtotheedgeswhoseend nodesareallmembersoffj 0 z+1 ;:::;j 0 n g.

Mutation. ThereareseveralpossibilitiesformutationonanacyclicorientationofG,butwehavedecided

to optforwhat seemsto bethe mostelementary onewhile guaranteeingthegeneration ofanotheracyclic

orientation. Givenanacyclicorientation! andthemutationpointi2N,theoperationconsistsofturning

node iinto asourceto yieldaneworientation! 0

. That! 0

isalso acyclichasbeenargued elsewhere(e.g.,

[4]), and the argument goes asfollows. If ! 0

is not acyclic, then the directed cycle that exists in it must

involvenodei, which iswheretheonlychangewaseectedon! toyield! 0

. Butthisisclearlyimpossible,

sinceaccordingto! 0

nodeiis asource.

3. The second formulation

Our second evolutionary formulationof thegraph coloringproblem is conceptuallysimpler thanthe rst.

Theindividualsthatitevolvesarepermutationsofthesequenceh1;:::;ni,inwhicheachnumberisusedas

anindex intoa certainreference sequenceof the nodesof G. Welet D =hi

1 ;:::;i

n

numbersforthepurposeofbreakingsymmetrywhenmorethanonenodewiththesamedegreeexist. InD,

i

x

comestotheleft ofi

y

ifeither thedegreeofi

x

islargerthanthedegreeof i

y

orthetwodegreesarethe

samebut i

x <i

y .

Thus, recalling that K(n) stands for the set of all permutations of h1;:::;ni, a permutation  =

hk

1 ;:::;k

n

i 2 K(n) is to be interpreted asa sequence of n steps to assign colors to the nodes of G. If

we let node(k)denote thekth node in sequence D (i.e., node(k) = i

k

), then the zth step in  assigns to

node(k

z

)thesmallestcolorthathasnotbeenassignedtoanyofitsneighborsinG. Ofcourse,theoutcome

ofsuch asequenceofstepsdependsonourchoiceonhowtobreaksymmetry;inSection6,wecommenton

thepossibleeectsofrandomnessandofchoosing analtogetherdierentreferencesequence.

Each  2 K(n) can in principle be regarded as a program to provide any graph on n nodes with a

coloring. Thegoalis to evolveonesuchprogram that, on average,canuseassmall anumberof colorsas

possible. However,thenumberof nonisomorphicgraphsonn nodesgrowssofastwith n(over165billion

graphs for n as small as 12 [31]) that it is not reasonable to expect acceptable performance oversuch a

wideclass of graphsof aprogramevolvedwithin tolerabletime bounds. Onthe otherhand,it also seems

improperto proceedas with therst formulationand evolveaprogram that is specic to asinglegraph,

sincenowtheverynature ofourindividualscalls forgreatergenerality.

Ourchoicehasbeentoevolveprogramsthatarespecictographsbelongingtoacertainclassofgraphs,

allhavingincommonnotonlythenumberofnodesbutalsothedensity,whichisdenedtobetheratioof

thegraph'snumberofedgesto themaximumpossiblenumberofedgeswhentherecanonlybeatmostone

edgebetweenanytwonodes. Weletpdenotesuchdensity,whichisthensuchthat

p= 2m

n(n 1) :

The class of graphs having n nodes and density psuch that p p  p + for 0  p  p + 1 will be denoted byG(n;p ;p + ).

Wenowdescribehowtoassessthetnessofanindividual,andhowtoperformcrossoveranddomutation

andinversion.

Fitness evaluation. Let T, henceforth referredto as the training set, be arandomly selected subset of

G(n;p ;p +

). Thetnessofanindividual2K(n)isbasedontheaveragenumberofcolorsthatindividual

requires overall membersof T. Ifforprogram we letcolors(;G) denote thenumberofcolorsrequired

forto assigncolorsto thenodesofgraphG, thenthetnessof is thenonnegativeintegerf

2 (), given by f 2 ()=n 1 jTj X G2T colors(;G): (5)

ProgramiscapableofcoloringthenodesofallgraphsinT with,onaverage,n f

2

()colors.

Crossover. Thecrossoveroftwoparentprograms 

1 ;

2

2K(n) toyield thetwoospring 0 1 ; 0 2 2K(n),

giventhecrossoverpointzsuchthat1z<n,worksasfollows. Let

 1 =hk 1 ;:::;k n i;  2 =h` 1 ;:::;` n i;  0 1 =hk 0 1 ;:::;k 0 n i;

 0 2 =h` 0 1 ;:::;` 0 n i: Then hk 0 1 ;:::;k 0 z i = hk 1 ;:::;k z i and hk 0 z+1 ;:::;k 0 n i is the subsequence of h` 1 ;:::;` n

i comprising all the

indices that are notalready in hk

1 ;:::;k z i. As for  0 2 , h` 0 1 ;:::;` 0 z i = h` 1 ;:::;` z i and h` 0 z+1 ;:::;` 0 n i is the subsequenceof hk 1 ;:::;k n

icomprisingalltheindicesthatarenotalreadyinh`

1 ;:::;`

z i.

Inheritance in this case is easier to identify than in the formulation of Section 2. Program  0 1 visits node(k 0 1 );:::;node(k 0 z

)forcoloringin thesamerelativeorder asprogram 

1 andnode(k 0 z+1 );:::;node(k 0 n )

inthesamerelativeorderas

2 . Likewise,program 0 2 visitsnode(` 0 1 );:::;node(` 0 z

)forcoloringinthesame

relativeorderasprogram

2 andnode(` 0 z+1 );:::;node(` 0 n

)inthesamerelativeorder as

1 .

Mutation. Single-locusmutationsaremeaninglessonaprogram2K(n),so whatwecallamutation in

thissecondformulationisreallyaswapoftwooftheindicesin,whichthenacquireeachother'spositions

inthesequence. Formutationpointsz andz 0

,theeectofthisoperationon=hk

1 ;:::;k n iistoyieldthe program 0 =hk 0 1 ;:::;k 0 n iwithk 0 z =k z 0 andk 0 z 0 =k z . Inversion. For program  =hk 1 ;:::;k n

i in K(n), the eect of inversion on to yield another program

 0 =hk 0 1 ;:::;k 0 n iistoset k 0 z =k n z+1 forz=1;:::;n.

4. On the evolutionary operators

Theevolutionaryoperatorsdescribedin Sections2and3foruserespectivelyin ourrstandsecond

formu-lationsofthegraphcoloringproblemwereselectedwithavarietyofgoalsinmind,includingmeaningfulness

in thecontext oftheformulationat hand,parsimonyinthe nalnumberofoperators,and expected

eec-tiveness (as faras this can be inferred from afew initial experimentson reasonably-sizedinstancesof the

problem). While it is possible to see relativelyeasily, as we argued in those sections, that the crossover

operator is indeed in bothformulationsresponsiblefor thetransmission to ospring ofstructurally

mean-ingful informationfrom theparents, for theremaining operators(mutation and, in thecase ofthe second

formulation,inversionaswell)itmayseemthat theyonlyfulll theelusiveroleof occasionallyhelpingthe

search escapelocal optima[17, 32]. In thissection, wearguethat mutation, in bothformulations, canbe

ascribed the more precise role of allowing the search to lead from any one individual to any other with

nonzeroprobability(possiblywiththehelpofinversion,inthecaseofthesecondformulation).

Understand-ingthisproperty,whichholdstriviallyinthecaseofbit-stringrepresentationsundersingle-locusmutations,

forexample,requiresalittleelaborationin thecaseofourtwoformulations,especiallysofortherstone.

WerstdiscussmutationinthecontextofSection2. Inthiscase,mutationonanacyclicorientationis

theturningofanarbitrarynode(themutationpoint)intoasource,whichyieldsanotheracyclicorientation.

Forarbitrary!;! 0

2(G), thequestionthatweansweraÆrmativelyinthissectioniswhetherthere exists

anite sequenceof mutations that turns! into! 0

. Theargumentrequires alittleadditionalnotation, as

follows. LetM E betheset of edgesthat ! and! 0

direct dierently, andlet U N betheset of end

nodesoftheedgesinM. Likewise,letM +

Ebethesetofalledgeslyingondirectedpathsaccordingto!

thatleadtonodesinU,andletU +

N bethesetofendnodesoftheedgesinM + . Clearly,U U + and MM +

. Thedesiredtransformationof!into ! 0

mustreversethedirectionsassignedby! toalledgesin

M whilekeepingalltheremainingedges' directionsuntouched.

Inorder todescribeanite sequenceof mutations thatachievesthis weneed stillmorenotation. For

k2N,lett

k

sequence ofmutations that alwaysguaranteethe transformationof ! into ! 0

. In this assignment,werst

lett

k

=0foreveryksuchthatk2NnU +

(here,andhenceforth,nisusedtodenotesetdierence). Asfor

k2U +

,weletvaluesbeassignedinsuchawaythat

t i =t j ; if(i;j)2=M; t i <t j ; if(i;j)2M (6) for(i;j)2M +

directed by! from ito j. What(6)isdoingis toassign valuesthatare strictlyincreasing

asonetraversesedgesofM alongthedirectionsgivenby!,butremainconstantoverthetraversalofedges

ofM +

nM alsoalongthedirectionsgivenby!.

Notethatsuchanassignmentisalwayspossible,owingtothefactthat! isacyclic,whichimpliesthat

the strict inequalities appearing in (6) for membersof M can always be satised. In particular, because

values are nondecreasing along directed paths, it is a trivial matter to choose them in such a way that

t k <jU + jforallk2U + . If ft k ;k 2 U +

g is aset of values that obeys (6), then the corresponding sequence of mutations that

leads from ! to ! 0

is obtainedby repeating the following step until t

k =0 for all k 2 U + . If ` 2 U + is

notcurrentlyasourceandt

`

isnonzeroandgreatestoverallofU +

,then turn`intoasourceandsett

` to

t

`

1;whentwoneighborsaretied,picktheonethatispointedto bytheedgebetweenthemaccordingto

thecurrentorientation. Notethat,ifeveryt

k

isassigned thesmallestpossiblevalue,thenthis sequenceof

mutationscomprisesO jU + j 2
mutationsoverall.

Nowlet e=(i;j)beany edge,and letthedirection assigned toit by! befrom i to j. Ife isnotan

edge of M +

, then clearlythe process we just described doesnot aect it. If e2 M +

nM, then t

i and t

j

haveinitially thesamevalue,and thereforei andj getturned into sourcesthesamenumberof timesand

alternately with respect toeach other. Also,becausee is directed towardj by!, j is therst ofthe two

nodestobeturnedintoasourceandi isthelast,soattheendtheedge'sdirectionsettlesasinitially, that

is, from i to j. Fore 2 M, initially it holds that t

i <t

j

, soj is turned into asource strictly moretimes

thaniis. Twoscenarioscanbeenvisaged. Intherst,theinequalityt

i <t

j

ismaintainedthroughoutthe

process,sojisthelastofthetwonodestobeturnedintoasource. Inthesecondscenario,t

i andt

j

become

equalbeforetheendoftheprocess,whichmusthappenupononeoftheturningsofj intoasource,therefore

leavingtheedgedirectedtowardi. From thenonthetwonodesbecomesourcesthesamenumberoftimes

and alternately,i beingthe rstand j thelast. Ineither scenario,edge eends updirectedfrom j toi. It

followsfromthis that! 0

istheorientationofthegraphattheendofthesequenceofmutations.

ThecaseoftheformulationofSection3isconsiderablysimpler. Fortwoprograms=hk

1 ;:::;k n iand  0 =hk 0 1 ;:::;k 0 n

i,thefollowingisasequenceof mutationsthat leadsfrom to  0 . For z=1;:::;n, check whetherk 0 z =k z

;ifnot,thenletz 0 >zbesuchthatk 0 z =k z 0 andswapk z withk z 0

. Clearly,theresulting

isthesameas 0

. Also,fastthoughthis transformationofinto  0

is(nomorethann 1swapsareever

needed),itiseasytoconceiveofcasesin whichtheuseofinversioncanmakeitevenfasterwhenappliedas

arststep.

5. The experimental test set

This section contains a brief description of the graphs to be used in Section 6 as benchmark graphs for

implementations of our rst and second formulations. They have all been extracted from the DIMACS

unorderedpairofnodesbyanedgewithprobabilityq=10. Weusethesegraphswithn2f125;250;500gand

q=5.

Rn.q. These arerandomgraphson nnodes. Theyareconstructedbyrandomlyplacingthenodesinsidea

unit squareand thenconnectingbyanedge anytwonodesthat arenofartherapartfrom each otherthan

q=10. Weusethesegraphswithn2f125;250gandq2f1;5g.

Rn.qc. These graphsare the complements of the Rn.q graphs, that is, each one of them has exactlythe

edgesthatareabsentfrom itscounterpart. Weusethesegraphswithn2f125;250gandq=1.

DSJRn.q[25]. ThesameasRn.q. Weusen=500and q=1.

DSJRn.qc[25]. ThesameasRn.qc. Weusen=500andq=1.

flatnk f [11]. Thesearek-partite randomgraphsonnnodeswithdensityp0:5. Beingk-partitemeans

thatthenodesetcanbepartitionedintokindependentsets,thatis,setswhosemembersarenotneighbors;

consequently, (G)k. Byconstruction,all independent sets haveas closeto thesamenumberof nodes

aspossible,andfurthermoreedgesaredistributedamongpairsofindependentsetsasequitablyaspossible.

Inaddition,givenanytwooftheindependentsets,nonodein oneofthem isallowedmoreedges joiningit

to theother setthan anyothernodeinitsownset (exceptfor anumberf of edges,known asa\ atness"

parameter). Weusethesegraphswithn=300,k2f20;26;28g,andf =0.

lenkx[29]. These arerandom graphson nnodes with(G)=k. Theyhavenumbersof edgessuch that

p,thedensity, issuchthat p0:25,andare constructedbythecreationofcliquesof varyingsizes insuch

awaythatthepre-speciedvaluek of(G)isnotviolated. Weusethesegraphswith n=450andk=15;

xisoneofa,b,c,ord,and isusedtodierentiateamonginstances.

mulsol.i.1[30]. Thisgraphresultsfromaninstanceoftheproblemofregisterallocationin compilers.

school1-nsh[30]. This graphresultsfrom theproblemofconstructingclass schedulesin such awayas to

avoidcon icts. Becauseitcomesfromanactualclassschedule,itisknownthat (G)14.

WeshowinTable1allthegraphsweemployinourexperiments. ForeachgraphG,thetabledisplays

thevaluesofn,m,andp,aswellas(G)(oranupperboundonit),whenknownfromdesigncharacteristics.

Thetable alsocontainsthegraphclassof which itis amemberthat istargeted inourexperimentsbythe

implementationofoursecond formulation.

6. Experimental results

Henceforth, we let Evolve AO and Evolve P denote the algorithms resulting, respectively, from the

evolutionaryformulationsofSections2and3. Evolve AOworksonaxedgraphGandseekstheacyclic

orientationofGwhoselengthiestdirected pathisshortestover(G). Evolve P,bycontrast,searchesfor

theprogram(amemberofK(n))thatcolorseachofthegraphsinG(n;p ;p +

)withasfewcolorsaspossible.

Both Evolve AO and Evolve P iterate for g generations, each one characterized by a population

G n m p (G) Targetclass R125.1 125 209 0:027 G(125;0:02;0:09) R125.5 125 3;838 0:495 G(125;0:40;0:99) DSJC125.5 125 3;891 0:502 G(125;0:40;0:99) R125.1c 125 7;501 0:968 G(125;0:40;0:99) mulsol.i.1 197 3;925 0:203 G(197;0:10;0:39) R250.1 250 867 0:028 G(250;0:02;0:09) R250.5 250 14;849 0:477 G(250;0:40;0:99) DSJC250.5 250 15;668 0:503 G(250;0:40;0:99) R250.1c 250 30;227 0:971 G(250;0:40;0:99) flat30020 0 300 21;375 0:477 20 G(300;0:40;0:99) flat30026 0 300 21;633 0:482 26 G(300;0:40;0:99) flat30028 0 300 21;695 0:484 28 G(300;0:40;0:99) school1-nsh 352 14;612 0:237 14 G(352;0:10;0:39) le45015a 450 8;168 0:081 15 G(450;0:02;0:09) le45015b 450 8;169 0:081 15 G(450;0:02;0:09) le45015c 450 16;680 0:165 15 G(450;0:10;0:39) le45015d 450 16;750 0:166 15 G(450;0:10;0:39) DSJR500.1 500 3;555 0:028 G(500;0:02;0:09) DSJC500.5 500 62;624 0:502 G(500;0:40;0:99) DSJR500.1c 500 121;275 0:972 G(500;0:40;0:99)

during all the evolutionary search, denoted respectivelyby ! 

and  

. For k > 1, the kth population is

generatedfrom thek 1stpopulationbyrsttransferring thefsttestindividuals tothenewpopulation

(this is an elitist rst step, with 0  f < 1), and then repeatedly selectingindividuals for application of

theevolutionaryoperators. Theresultingindividualsarethenaddedtothenewpopulationuntilitislled.

Therstpopulationisgenerated randomly,whichcanbeachievedin asimilar fashionbybothalgorithms

byrandomlycreatingssequencesofnintegers. Ineachofthesesequences,everyelementoff1;:::;ngmust

appear exactlyonce. ForEvolve AO,one such sequenceis identiedwith thelinearrepresentationL(!)

ofsomeacyclicorientation!;forEvolve P,itisidentiedwithaprogramdirectly.

As with the choice of evolutionary operators for both algorithms, choosing an appropriate selection

method has relied on the outcome of some preliminary experiments on reasonably-sized graphs. For

Evolve AO, this hasresulted in selecting individuals proportionally to its linearly normalizedtness in

thecurrentpopulation. Inotherwords,iffork2f1;:::;sgacertainindividual! hasthekthsmallestvalue

off

1

(!)overtheentirepopulationforf

1

(!)asin(4), thenitisselectedwithprobabilityproportionalto

g

1

(!)=2k: (7)

Tiesin theorderof tnesswithin thepopulationarebrokenby theorder inwhich individualswere added

tothepopulationin therstplace.

ThecaseofEvolve Pissimilar, but becausetnessesare nowaveragestakenoverthemembersofa

subclassofgraphsT ofG(n;p ;p +

proportionally to afunction that verysteeply increaseswith f

2

() for  in the current population, f

2 ()

beinggivenasin(5). Specically,weletbeselectedwithprobabilityproportionalto

g

2 ()=e

f2()

: (8)

The decision as to how to manipulate the individuals selected according to either (7) or (8) before

inclusion in the new population is also reached probabilistically. The various probabilities involved are

denoted asfollows.

p

c

: theprobabilityof performingthecrossoveroftwoselectedindividuals;

p

m

: theprobabilityofperformingmutation onaselectedindividual;

p

i

: theprobabilityofperforminginversiononaselectedindividual.

Inthissection, wesummarizetheresultsofourexperimentswithEvolve AOand Evolve Ponthe

benchmarkgraphsdescribedin Section 5. Thesummarywepresentis farfrom exhaustivewithrespect to

thepossiblecombinationsoftheparametersinvolved,butrathercontainsthecombinationsthatyieldedthe

bestresultswehaveobtained.

Wepresenttwotypesofperformancegures. Figuresofthersttypere ecthowwellagraphorclass

of graphsiscolored byagiven individual. ForEvolve AO,which operates onxed G,this performance

gure is the numberof colorsneeded to provide G with a coloring. During the evolutionary search, this

number is given as n f 1 (! + ), where ! +

is the best individual of the current generation; likewise, once

! 

hasbeenidentied asthebest individualfoundduring thewholesearch,then itisgivenas n f

1 (!



).

ForEvolve P,whichaimsatcoloringallthegraphsbelongingtotheclassG(n;p ;p +

)bytheprogram 

thatonaverageperformsbestonthetrainingsetT G(n;p ;p +

),thisrstperformancegureispresented

undertwoguises. Duringtheevolutionarysearch,itispresentedastheaveragenumberofcolorsneededby

thegraphsinT underthebestprogram,say +

,ofthecurrentgeneration,thatis,n f

2 ( + ). After  has

beenidentied, then the performance gureis presentedascolors( 

;G) for eachgraph G2G(n;p ;p +

)

thatisin thebenchmarkset asintroducedinSection 5.

Oursecondperformancegureisrelatedtohowthequalityofthebestindividualoutputbythesearch

is aected by thenumber g of generations elapsedbefore completion. This gureis givendirectly by the

valueofg,whichisto bethoughtofasaplatform-independentassessmentofeach algorithm'seÆciency.

This section also contains a comparativeassessment of Evolve AO and Evolve P with respect to

seven other approachesthat havealso been tested on theDIMACS benchmark graphs. These are thesix

approaches that appeared in the challenge's proceedings volume [26] and the one in [18], which appeared

later. Becausenearlyallnineapproachesarebasedonwidelydieringstrategiesandhavebeenimplemented

onanequallydiversesetofplatforms,ourcomparisonisrestrictedtoperformanceguresthatindicatehow

wellthebenchmarkgraphsarecolored. Thefollowingisabriefdescriptionofthosesevenapproaches.

I Greedy [11]. Thisis aniteratedversionofthegreedyprocedurethat assignscolorstonodesfollowinga

certain orderand choosing,for each node, the smallestcolorthat still hasnot been assignedto anyof its

neighbors. Ateachiteration,adierentorderischosenaccordingtoacriterionthatinvolvesthelatestcolor

assignmentandalsodegree-relatedinformation, asinDSatur, amongothers.

T B&B[23]. This isabranch-and-boundalgorithm thatemployselementsinspiredin tabusearch [22]for

isgoodforeitherexactorheuristicgraphcoloring,dependingonwhetherthelargecliquethatisfoundcan

beguaranteedtobemaximum(havemaximumsize),sinceitcanbeeasilyshownthat(G)!(G), where

!(G)isthesizeofthemaximumcliqueofG[7]. Providingsuchaguaranteeisofcourseashardascoloring

thegraph[20,27], butseemsto becomeonlytolerablyburdensomeasthegraphsbecomesparser.

Dist [34]. This algorithm employs multiple sequential computers to work concurrently on several partial

coloringsof thegraph(that is, assignmentsof colorsto subsets ofN). Oneof thecomputersmaintainsa

poolofthebestpartialcoloringssofaroutputbytheothers,andcontinuallysubmitssuchcoloringstothose

computersto beextendedbyneighborhoodsearchand possiblyimproved.

Par [30]. This is a combination of Dist with exhaustive search by parallel branch-and-bound. The two

computationsarestartedconcurrentlyonaparallelmachine,eachonetakingacertainnumberofprocessors.

Theprocessorthatin Distisresponsibleformaintainingapoolofpartial coloringsis nowalsofed bythe

exhaustive-searchcomputation,andrelaysthebestpartialcoloringsfoundbyonecomputationtotheother.

E B&B[36]. Thisisanexactbranch-and-boundalgorithm. Itskeyingredientsincludendingamaximum

cliqueto work aslowerbound (cf. ourearlierdescriptionofT B&B) andanovelbranchingrule that,like

DSatur,usesthenodes'saturationdegreesinthedecision.

T Gen 1 [16]. This is a hybrid approach that employs tabu search inside a genetic algorithm. In this

approach, anindividual isapartition ofthegraph'snodesinto colorclasses,that is,subsetsofnodesallof

whicharetobeassignedthesamecolor. Thegenerationofnewindividualsbycrossoverdoesnotinprinciple

guaranteetheabsence ofedges joining nodes in thesamecolorclass, which is where tabusearch comes in

lookingforthere-arrangementinto colorclassesthatrequirestheleastnumberofclasses.

T Gen 2[18].ThisapproachisentirelyanalogoustoT Gen 1,fromwhichitdiersmainlyonhowcrossover

isperformed.

The plots in Figure 1 show the behavior of Evolve AO, each plot corresponding to a run of the

algorithmononeofthegraphsinTable1. Theyhaveallbeenobtainedwithp

c

=0:60,p

m

=0:02,f =0:70,

g=5;000,ands=100. Forclarityofexposition,thealgorithm'sbehaviorpriortothehundredthgeneration

isnotshown(attherstgeneration,andshortlyonward,thenumberofcolorsimpliedbythebestindividual

tendstobeinordinatelyhigh). Analogously,Figure2containsoneplotforeachoftheelevendistincttarget

classesappearinginTable1,eachplotcorrespondingtoarunofEvolve Ponrandomlyselectedmembers

ofitstargetclass. Theseplotshavebeenobtainedwithparametersvaluedasin Table2.

The plots in Figures 1 and 2 indicate that both Evolve AO and Evolve P exhibit the

tness-improvement behaviorone expects of well formulated evolutionary computations, often verymarkedly so

in the earliestgenerations. Whilefor Evolve AOthis happens smoothly overtime, forEvolve P some

plotsarealittlejagged,perhapsre ectingthegreaterdiÆcultyofconcomitantlyimprovingthecoloringsof

allgraphsinthetrainingsetT.

Weshowin Table3the best resultsobtainedby Evolve AOandEvolve P side bysidewith those

obtainedby thecompeting algorithms we outlinedearlier. Foreachof the graphsin Table1, what Table

3showsis the value of itschromaticnumber (or an upperbound on it), when known by design, and the

0

1

2

3

4

5

6

100 500 1000

2000

5000

Coloring

Generation

34

35

36

37

38

39

40

41

42

100 500 1000

2000

5000

Coloring

Generation

15

20

25

30

35

40

100 500 1000

2000

5000

Coloring

Generation

40

45

50

55

60

65

70

75

80

100 500 1000

2000

5000

Coloring

Generation

R125.1 R125.5 DSJC125.5 R125.1c

20

25

30

35

40

45

50

55

60

100 500 1000

2000

5000

Coloring

Generation

6

8

10

12

14

16

100 500 1000

2000

5000

Coloring

Generation

60

65

70

75

80

100 500 1000

2000

5000

Coloring

Generation

28

30

32

34

36

38

100 500 1000

2000

5000

Coloring

Generation

mulsol.i.1 R250.1 R250.5 DSJC250.5

60

65

70

75

80

85

90

100 500 1000

2000

5000

Coloring

Generation

20

30

40

50

60

70

80

90

100 500 1000

2000

5000

Coloring

Generation

30

40

50

60

70

80

90

100

110

100 500 1000

2000

5000

Coloring

Generation

30

40

50

60

70

80

90

100

100 500 1000

2000

5000

Coloring

Generation

R250.1c flat300 200 flat300 260 flat30028 0

10

15

20

25

30

35

40

100 500 1000

2000

5000

Coloring

Generation

15

20

25

30

100 500 1000

2000

5000

Coloring

Generation

15

20

25

30

100 500 1000

2000

5000

Coloring

Generation

15

20

25

30

35

40

45

100 500 1000

2000

5000

Coloring

Generation

school1-nsh le45015a le45015b le45015c

15

20

25

30

35

40

45

100 500 1000

2000

5000

Coloring

Generation

10

12

14

16

18

20

22

24

100 500 1000

2000

5000

Coloring

Generation

50

55

60

65

100 500 1000

2000

5000

Coloring

Generation

84

86

88

90

92

94

100 500 1000

2000

5000

Coloring

Generation

le45015d DSJR500.1 DSJC500.5 DSJR500.1c

Figure1. ConvergenceofEvolve AOonthegraphsofTable1

is given as an average overa certain number of runs (10 for I Greedy, 5for Par, between 3and 10 for

T gen 1, and 5 for Evolve AO), for some others as the result of a single run (T B&B, E B&B, and

Evolve P). Yet forother algorithms(Distand TGen 2), what appearsin Table3isthelowestvalueof

6.22

6.24

6.26

6.28

6.30

6.32

6.34

6.36

6.38

6.40

0

10

20

30

40

50

60

70

Coloring

Generation

34.00

34.20

34.40

34.60

34.80

35.00

35.20

35.40

35.60

0

50

100

150

200

250

300

Coloring

Generation

18.20

18.40

18.60

18.80

19.00

19.20

19.40

0

20

40

60

80 100 120 140 160 180 200

Coloring

Generation

G(125;0:02;0:09) G(125;0:40;0:99) G(197;0:10;0:39)

8.75

8.80

8.85

8.90

8.95

9.00

9.05

9.10

9.15

9.20

9.25

0

50

100

150

200

250

300

350

Coloring

Generation

63.00

63.50

64.00

64.50

65.00

65.50

0

100

200

300

400

500

600

Coloring

Generation

57.80

58.00

58.20

58.40

58.60

58.80

59.00

59.20

59.40

59.60

59.80

0

100

200

300

400

500

600

700

Coloring

Generation

G(250;0:02;0:09) G(250;0:40;0:99) G(300;0:40;0:99)

27.80

28.00

28.20

28.40

28.60

28.80

29.00

0

100

200

300

400

500

600

Coloring

Generation

13.20

13.30

13.40

13.50

13.60

13.70

13.80

13.90

14.00

0

100 200 300 400 500 600 700 800 900

Coloring

Generation

33.80

33.90

34.00

34.10

34.20

34.30

34.40

34.50

0

50

100

150

200

250

300

Coloring

Generation

G(352;0:10;0:39) G(450;0:02;0:09) G(450;0:10;0:39)

15.70

15.75

15.80

15.85

15.90

15.95

16.00

16.05

16.10

0

50

100

150

200

250

300

350

400

Coloring

Generation

101.0

101.5

102.0

102.5

103.0

103.5

0

50

100

150

200

250

Coloring

Generation

G(500;0:02;0:09) G(500;0:40;0:99)

Figure 2. ConvergenceofEvolve PonthetargetclassesofTable1

testedbysuccessivelydecreasingthevalueofkfromsomeinitialvalue. TheentriesinTable3corresponding

toallalgorithmsbutourownareeither from[26]or,inthecaseofT Gen 2, from[18].

WhilenoneoftheheuristicsappearinginTable3canbeproclaimedstrictlybestoverallgraphs,when

we examineeach graphindividually many ofthe heuristicscanbesaid to doaswellas theonethat does

Targetclass jTj p c p m p i f g s G(125;0:02;0:09) 40 0:71 0:01 0:28 0:02 70 200 G(125;0:40;0:99) 60 0:71 0:01 0:28 0:02 300 100 G(197;0:10;0:39) 60 0:71 0:01 0:28 0:01 200 100 G(250;0:02;0:09) 60 0:71 0:01 0:28 0:01 350 100 G(250;0:40;0:99) 60 0:71 0:01 0:28 0:02 600 100 G(300;0:40;0:99) 60 0:71 0:01 0:28 0:01 700 100 G(352;0:10;0:39) 60 0:71 0:01 0:28 0:01 600 100 G(450;0:02;0:09) 100 0:71 0:01 0:28 0:02 900 100 G(450;0:10;0:39) 100 0:70 0:01 0:29 0:02 300 100 G(500;0:02;0:09) 40 0:70 0:01 0:29 0:02 400 100 G(500;0:40;0:99) 40 0:71 0:01 0:28 0:02 250 200

forwhichperformanceguresaregivenin bold typeface)performedat leastas wellasthebest performers

on those samegraphs, occasionallyeven better. Forthe other graphs, oftenEvolve AO and Evolve P

missthebestperformersverynarrowly.

7. Concluding remarks

Wehavein this paperintroducedtwonovelevolutionary formulationsof thegraphcoloringproblem. The

rst formulation views the problem of nding a graph's chromatic number as the problem of nding an

acyclicorientationof thegraphaccordingto which thelongestdirected pathis shortestamong allacyclic

orientations. Viewing theproblem fromthis perspectiveimmediatelyprovidestheessentialfoundation for

the evolutionary formulation, since each acyclic orientation can be viewed as an individual whose tness

canbesaidto behigherasitslongestdirectedpathis shorter. Despitethisinitial simplicity,thedesignof

appropriateevolutionaryoperatorshasreliedonsophisticatedgraph-theoreticarguments.

Our second formulation takesan entirelydierent route, and seeks to nd a program that will color

anygraphhavingapre-speciednumberofnodesanddensitywithinapre-speciedinterval. Thisprogram

isapermutation ofindicesinto asequenceofnodesthat ispreviouslyagreedupon. Thesequencewehave

usedcontainsnodesinnonincreasingorderofdegrees,havingbeenlooselyinspiredbytheDSaturheuristic.

Butbeingonlyareference sequence,anyother sequencecouldhavebeenused aswell. A programis then

somethinglike\rst colorthe nodewhose degreeis thethird highestwith thelowest availablecolor, then

the node whose degree is the seventh highest, etc.," for example. In this formulation, each individual is

aprogram, its tnessbeinghigher as the average numberof colorsit requires to colorall thegraphsin a

randomlyselectedsubsetofgraphshavingthatnumberofnodesanddensityin thatintervalissmaller.

We have presented experimental results on some of the DIMACS benchmark graphs and compared

ouralgorithms'performanceswiththose oftheotherheuristicsthat weknowtohavebeentestedonthose

graphsaswell. Ourresultsindicate generallycompetitiveperformance,sometimesreachingthebest results

obtainedbytheotherheuristics,occasionallybetter.

Notwithstanding these positive results, there is room, at least in principle, for improvements to be

G (G) I Greedy TB&B Dist Par E B&B R125.1 5:0 5 5 5:0 5 R125.5 36:9 36 36 37:0 36 DSJC125.5 18:9 20 17 17:0 R125.1c 46:0 46 46 46:0 46 mulsol.i.1 49:0 49 49 49:0 49 R250.1 8:0 8 8 8:0 8 R250.5 68:4 66 65 66:0 65 DSJC250.5 32:8 35 28 29:2 R250.1c 64:0 65 64 64:0 64 flat300 200 20 20:2 39 20 20:0 flat300 260 26 37:1 41 26 32:4 flat300 280 28 37:0 41 31 33:0 school1-nsh 14 14:1 26 20 14:0 le45015a 15 17:9 16 15 15:0 le45015b 15 17:9 15 15 15:0 15 le45015c 15 25:6 23 15 16:6 le45015d 15 25:8 23 15 16:8 DSJR500.1 12:0 12 12 12:0 12 DSJC500.5 58:6 65 49 53:0 DSJR500.1c 85:0 87 85 85:3

symmetry, so it may be worth checking whether introducing randomness at these points can have any

noticeableeect. Also,ourtwoformulationsarebothpurely evolutionary,inthesense that atnopointdo

theyemployauxiliaryheuristics,likeforexamplesomeformoflocalsearch. Wethinkitmaybeworthwhile

toinvestigatesuchhybridalternatives.

Itis,however,inthecontextofEvolve P,thealgorithmthatimplementsoursecondformulation,that

we believe the possibilities for extension and improvement are the most challengingand interesting. For

example,ofallthegraphsthatwerepresentedtoEvolve P forcoloring,theones thatpromptedtheleast

satisfactoryresultswerethosewhosenodeshavedegreesverycloselyconcentratedneartheaverage,thatis,

graphswithverylittlevarianceofnodedegree(thiscannotbeinferredfromthedatawehavepresented,but

isclearfromacloserexaminationoftheDIMACSles). Wethinkthismaybeduetothefactthatperhaps

thisquantitywaspoorlyrepresentedbythegraphsofthetrainingset. BecauseEvolve Pisstronglybased

ondegreesandhowtheyrelatetooneanotherinsidethegraph,itispossiblethattargetingtheevolutionnot

justatacertainnumberofnodesandacertaindensityinterval,butalsoatacertainintervalofnode-degree

variance,mightyield signicantimprovementsin somecases. Thiswould, ofcourse, callforageneratorof

randomgraphscapableofcontrollingsuchvariancesinadditiontodensities,whichistoourknowledgealso

G (G) T Gen 1 TGen 2 Evolve AO Evolve P R125.1 5:0 5:0 5 R125.5 35:6 36:2 36 DSJC125.5 17:0 17:2 20 R125.1c 46:0 46:0 46 mulsol.i.1 49:0 49:0 49 R250.1 8:0 8:0 8 R250.5 69:0 65:2 65 DSJC250.5 29:0 28 29:1 29 R250.1c 64:0 64:0 64 flat300 200 20 20:0 26:0 23 flat300 260 26 26:0 31:0 28 flat300 280 28 33:0 31 33:0 29 school1-nsh 14 14:0 14:0 20 le45015a 15 15:0 15:0 17 le45015b 15 15:0 15:0 17 le45015c 15 16:0 15 16:0 25 le45015d 15 16:0 19:0 25 DSJR500.1 12:0 12:0 12 DSJC500.5 51:0 48 52:5 59 DSJR500.1c 85:3 85:0 85 Acknowledgments

The authors acknowledge partial support from CNPq, CAPES, the PRONEX initiative of Brazil's MCT

undercontract41.96.0857.00,andaFAPERJBBPgrant.

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