An ACO Algorithm for. the Graph Coloring Problem

Transcription

1 Int. J. Contemp. Math. Scences, Vol. 3, 2008, no. 6, An ACO Algorthm for the Graph Colorng Problem Ehsan Salar and Kourosh Eshgh Department of Industral Engneerng Sharf Unversty of Technology, Tehran, Iran Abstract Ant Colony Optmzaton (ACO) s a well-nown metaheurstc n whch a colony of artfcal ants cooperates n explorng good solutons to a combnatoral optmzaton problem. In ths paper, an ACO algorthm s presented for the graph colorng problem. Ths ACO algorthm conforms to Max-Mn Ant System structure and explots a local search heurstc to mprove ts performance. Expermental results on DIMACS test nstances show mprovements over exstng ACO algorthms for the graph colorng problem. Mathematcs Subject Classfcaton: 05C5 Keywords: Graph colorng, meta-heurstc algorthms, ant colony optmzaton I. INTRODUCTION One of the most well-studed combnatoral optmzaton problems s the graph colorng problem. Gven an undrected graph G=(V, E), the problem s to fnd a colorng of the vertces wth mnmum number of colors such that no par of adjacent vertces has the same color. Graph colorng problem s expected to have a wde varety of applcatons such as schedulng [9, 7], frequency assgnment n cellular networs [2], tmetablng [6], crew assgnment [8], etc. Two classes of algorthms are avalable to solve ths problem: exact and approxmate algorthms. Snce t has been proved that the graph colorng problem belongs to the class of NP-hard problems [3], exact algorthms [2, 9, 22] are

2 294 E. Salar and K. Eshgh confned to solve small sze nstances and as the problem sze ncreases, the use of ths class of algorthms qucly becomes nfeasble. Therefore, the only possblty s to resort to approxmate algorthms n order to obtan near-optmal solutons at relatvely low computatonal costs. Approxmate algorthms for the graph colorng problem can be classfed nto three man classes: constructve heurstcs [5], local search heurstcs [3], and metaheurstcs. However, the majorty of the approxmate algorthms are due to metaheurstcs mplementatons. Several metaheurstcs are appled to graph colorng problem such as Smulated Annealng, Tabu Search, Genetc Algorthm, Ant Colony Optmzaton, etc. The frst attempt to apply Smulated Annealng metaheurstc to the graph colorng problem s due to Johnson et al. [6]. Two Tabu Search algorthms are suggested by Hao and Dorne [8] and Hertz and Werra [5]. Fleurent and Ferland [0] lewse Costa et al. [4] proposed two dfferent Genetc Algorthm metaheurstcs whch are the state-of-the-art algorthms for the graph colorng problem. Hao and Galner [] hybrd algorthm s yet another promsng concept for the graph colorng problem, t uses Tabu Search as well as Genetc Algorthm to color a graph. Ant Colony Optmzaton [7] mplementatons n the context of the graph colorng problem are those of Costa and Hertz [5] whch embed two graph colorng constructve heurstcs RLF [7] and DSATUR []. Expermental results of ther proposed algorthm, ANTCOL, were promsng but far behnd state-ofthe-art algorthms. In ths paper, a modfcaton of ANTCOL, a Max-Mn ant system [2] algorthm for Graph Colorng (MMGC), s proposed to mprove ANTCOL performance. Computatonal results on DIMACS test nstances [22] demonstrate that MMGC outperforms ANTCOL. The remander of ths paper s organzed as follows. In secton II, some defntons and notatons are presented. In secton III, an ANTCOL algorthm s brefly ntroduced. ANTCOL drawbacs as well as possble extensons are dscussed n secton IV. ANTCOL modfcaton, MMGC, s proposed n secton V and computatonal results are reported n secton VI. Fnally, the paper concludes n secton VII. II. PROBLEM DEFINITION Let G = ( V, E) be an undrected graph where V s the set of vertces and E s the set of edges. An ndependent set s a subset of vertces n whch no par of adjacent vertces exsts. A q-colorng of G s a mappng c : V {, 2, 3,..., q} that assgns colors to vertces. The colorng s feasble f no two adjacent vertces has he same color,.e. { u, v} E : c ( u) c ( v), otherwse conflcts happen. A colorng wth at least one conflct s called an nfeasble colorng. An optmal colorng of G s a feasble colorng wth smallest number of colors. Ths mnmum number of colors q for whch a feasble q-colorng exsts s called the chromatc number of G and s denoted by χ (G). Gven a graph G, the graph colorng problem s to fnd an optmal colorng. In addton, graph colorng can be formulated as a setparttonng problem; a feasble q-colorng s a parttonng of set V nto q ndependent sets,.e.

3 Graph colorng problem 295 q V = U C, j, j : C C = φ () = Independent sets are also called color classes. By analogy, the objectve s to partton the set V nto mnmum number of color classes. j III. ANTCOL ALGORITHM ANTCOL proposed by Costa and Hertz [5], s a metaheurstc to near-optmally solve the graph colorng problem. In ANTCOL a colony of artfcal ants teratvely colors a specfc graph, at each teraton, ntally, ants produce feasble colorngs by consderng pheromone trals and heurstc nformaton, and afterwards pheromone trals are updated accordng to the qualty of colorngs. The qualty of colorngs are measured usng the followng evaluaton functon, f ( s) = (2) q ( s) where q (s) denotes the number of colors appled n colorng s. Pheromone trals are related to pars of nonadjacent vertces. Therefore, each par of nonadjacent vertces { v, v j} has an assocated pheromone tral j that represents the colony experence of colorngs n whch the two mentoned vertces have the same color,.e. belong to the same color class. Artfcal ants produce feasble colorngs of the graph usng modfed versons of RLF and DSATUR, called ANTRLF and ANTDSATUR, respectvely. Snce ANTRLF outperforms ANTDSATUR, only the frst one s descrbed here. In ANTRLF, analogous to RLF, there exst several stages, at stage the artfcal ant constructs color classc, and stage also conssts of several steps, at each step the artfcal ant determnes whch uncolored vertex to be added to the color classc. Let W be the set of uncolored vertces that can be added toc, and B be the set of uncolored vertces whch are not allowed to be added toc. In order to choose uncolored vertex v, ANTRLF can use three dfferent peces of heurstc nformaton as follows η = deg B ( v ) (3) η = deg B W ( v ) (4) η = W degw ( v ) (5) Each heurstc nformaton defnton leads to a strategy n ANTRLF. However, at the begnnng of stage, there are no vertces n B, so the followng two strateges are appled to add the frst vertex toc. Randomly selectng an uncolored vertex from W. Selectng vertex v wth maxmum degw ( v ).

4 296 E. Salar and K. Eshgh Thus 6 dfferent strateges are obtanable n ANTRLF va combnng heurstc nformaton strateges and the two above ones. ANTRLF experments determned the selecton of (3) and "Randomly selectng an uncolored vertex from W" as the best strategy. Pheromone trals are ntally set to and at the end of each teraton, they become updated consderng the followng rule j = ( ρ) j + s S j q ( s) (6) where q (s) represents the number of colors appled to colorng s and ρ denotes the pheromone evaporaton rate. S j s the subset of colorngs n whch the two nonadjacent vertces v and v j belong to the same color class. In order to choose an uncolored vertex v to be added to the color classc, pheromone tral s defned as follows j j C = (7) C contans all the pheromone trals between vertex v and so far added vertces n color class C, n other words, t represents the colony experence of settlng vertex v wth other vertces of C n the same color class. Consequently, at each step of stage, the probablstc decson rule determnes whch uncolored vertex v W to be added to the color class C as follows: α β η v W α β P = j η j (8) j W 0 v W where p s the probablty of selectng vertex v. Stage contnues whle W remans nonempty. Adjusted parameters for ANTRLF are α = 2, β = 4 and ρ = The colony sze s set to 00 ants and termnaton condton s defned as the number of teratons exceeds 50. ANTRLF appled to 4 samples of random graphs, expermental results were satsfactory but consderably outperformed by state-of-the-art algorthms. IV. MOTIVATIONS There are some drawbacs n ANTCOL whch dmnsh ts performance. In the followng, these drawbacs are outlned. Although ANTCOL explots an ngenous pheromone tral defnton, ts pheromone updatng rule (6) seems qute neffcent; regardng the fact that ateach teraton all ants depost pheromone, ANTCOL rapdly converges towards medocre colorngs. Generally speang, all ACO algorthms whch conform to Ant System structure suffer from stagnaton. Subsequent ACO structures try to resolve ths problem.

5 Graph colorng problem 297 ANTCOL evaluaton functon (2) s not a sutable measure of colorngs due to exstence of a wde range of colorngs wth the same number of colors [9]. For example let s and s 2 be two colorngs of the specfc graph wth the followng characterstcs; they have equal number of color classes but the frst one has a color class n whch there exsts only one vertex, whereas the second one has color classes n whch vertces are evenly dstrbuted, ntutvely, s seems more promsng than s 2, but ANTCOL s ncapable of dstngushng between them. Consequently, ths leads to neffectve explotaton of explored colorngs. Most of the ACO algorthms utlze a local search heurstc to mprove the obtaned solutons at each teraton, Nevertheless, ANTCOL lacs an effcent local search whch can contrbute to better exploraton of the soluton space. Fnally, ANTCOL uses the probablstc decson rule (8) to select an uncolored vertex to be added to the color class under constructon. However, at the begnnng of each stage, t uses a "randomly choosng" strategy that does not tae advantage of nether pheromone trals nor heurstc nformaton whle the color class under constructon s notceably affected by the frst vertex chosen. Consequently, the frst vertex should be selected more delberately. Applyng approprate heurstc nformaton n selectng the frst vertex may result n better performance of ANTCOL. MMGC s a modfcaton of ANTCOL wth the am of obvatng the above drawbacs. V. MMGC ALGORITHM Our proposed ACO algorthm, MMGC, conforms to Max-Mn ant system structure. At each teraton a colony of artfcal ants colors a specfc graph, and afterwards only the teraton-best ant deposts pheromone accordng to the qualty of ts colorng. In the followng subsectons, MMGC detals are descrbed. A. Colorng the Graph Each artfcal ant colors the vertces of the graph usng a modfed verson of ANTRLF. By analogy, ts colorng conssts of several stages. At each stage, a color class s bult up. Each stage contans several steps. At each step, the artfcal ant determnes the uncolored vertex whch to be added to the color class under constructon usng the probablstc decson rule (8). Pheromone trals are related to pars of nonadjacent vertces, the same as ANTCOL. Heurstc nformaton (3) asssts the artfcal ant to select the proper uncolored vertex at each step. However, snce at the begnnng of stage, color class C s empty, a new probablstc decson rule s defned to add the frst vertex. Ths rule s just a functon of the followng heurstc nformaton. η = degw ( v ) (9) Hence the assocated probablstc decson rule s as follows:

6 298 E. Salar and K. Eshgh β η v W β P = η j (0) j W 0 v W where p denotes the probablty of selectng vertex v as the frst vertex of C. B. Evaluatng the Colorngs Johnson et al. n [6] proposed a mnmzaton evaluaton functon for graph colorng problem, as follows q = q f ( s) = C + 2 C E( C ) () 2 = n whch E ( C ) represents the set of conflcts n the color class C. Ths evaluaton functon conssts of two parts, as one tres to mnmze t through an exploraton; the frst part bases the exploraton towards colorngs wth fewer and bgger color classes and the second part towards colorngs wth fewer conflcts. They also showed that Local mnmum of evaluaton functon () were pertnent to feasble colorngs. Regardng a soluton space comprses of only feasble colorngs, the second part s elmnated and evaluaton functon () becomes: q = 2 f ( s) = (2) C Evaluaton functon (2) can wdely demonstrate the dfferences between feasble colorngs. However, smaller values of (2) do not necessarly result n fewer color classes, t bases towards unevenly dstrbuted colorngs rather than optmal ones, and therefore t can not solely gude the exploraton towards near-optmal colorngs. In order to prevent the mentoned dsadvantage, MMGC explots number of color classes as well as the followng maxmzaton evaluaton functon to evaluate dfferent colorngs q 2 C = f ( s) = (3) At each teraton, ants colorngs are prortzed accordng to the non-decreasng order of the number of color classes, and tes are broen accordng to the nonncreasng order of evaluaton functon (3). Iteraton-best ant s the one wth pror colorng. C. Pheromone Tral Intalzaton and Lmts Consderng Max-Mn ant system structure, Pheromone trals are ntalzed to 2 V / ρ, also, MMGC lmts pheromone trals to nterval [ mn, max ], where and mn are computed accordng to the followng equatons max

7 Graph colorng problem 299 gb f ( s ) max = ρ (4) max mn = a (5) ( avg ) n Pbest a = n Pbest (6) n whch, s gb denotes the global-best soluton, and at the end of each teraton, pheromone tral lmts are updated accordng to the so-far obtaned s gb. Snce, on the average, there exst V / 2 uncolored vertces to be selected at each step, the parameter avg equals to V / 2 n (6) and P best represents the probablty of constructng the convergence colorng. D. Applyng Kempe Chan Local Search At each teraton, MMGC mproves Iteraton-best colorng usng Kempe chan local search heurstc [20]. The local search starts wth an ntal soluton correspondng to teraton-best colorng and explores through Kempe chan neghborhood structure usng a frst mprovement strategy. It uses the evaluaton functon (3) and holds the feasblty of colorngs durng exploraton. E. Pheromone Trals Updatng Rule In MMGC, only the teraton-best ant deposts pheromone consderng the followng pheromone updatng rule q j = ( ρ) j + C, j : v v C q, j :,2,..., (7) = 2 In other words, regardng the teraton-best colorng, each pheromone tral, assocated wth a par of nonadjacent vertces n the same color class, receves pheromone proportonal to evaluaton functon (3). VI. COMPUTATIONAL RESULTS Costa and Hertz appled ANTCOL to Fleurent and Ferland's randomly generated graphs and reported ther results. We appled MMGC to DIMACS test nstances and n order to obtan a far comparson, we also reran ANTCOL on these test nstances. Both algorthms were coded n Delph 7 and run on a Pentum 4 PC wth.8ghz CPU and 256MB RAM. Our expermental report conssts of two parts; frst, MMGC and ANTCOL runnng behavors on G random 250,0. 5 graph are presented, and then, the results of comparng MMGC to ANTCOL on other test nstances are reported. A. MMGC and ANTCOL Runnng Behavors We ran MMGC as well as ANTCOL on G. Adjusted parameters for 250,0. 5 MMGC were as follows, α = 2, β = 4, β = 3, ρ = 0. 04, and P = The colony best

8 300 E. Salar and K. Eshgh sze was set to 20 ants and the termnaton condton was defned as the number of teratons exceeds 300. ANTCOL parameters were the same as Costa and Hertz's adjustments. Runnng results lead to the followng results. MMGC taes longer tme to explore the soluton space whereas ANTCOL rapdly converges towards medocre colorngs. The followng plot depcts the MMGC and ANTCOL convergence behavors. Fg.. The Convergence Behavor of ANTCOL and MMGC Besdes, λ -Branchng Factor can clearly show the pheromone dstrbuton durng MMGC and ANTCOL teratons. The followng plot also verfes the long MMGC exploraton phase. Fg. 2. λ -Branchng Factor n ANTCOL and MMGC Iteratons ( λ = 0.05) In MMGC, Kempe chan local search heurstc s appled to mprove the teratonbest colorngs. Ths local search ntally causes dramatc changes to colorngs, but afterwards, t becomes less effectve. Kempe chan may even reduce the number of color classes n the prmary trals. The plot below, llustrates the teraton-best evaluaton functon (3) value before and after applyng Kempe chan local search heurstc.

9 Graph colorng problem 30 Fg. 3. Improvng the Evaluaton Functon usng Kempe Chan Local Search B. MMGC and ANTCOL Comparson MMGC has been compared to ANTCOL usng random graphs, Leghton, Queen and Class Schedulng graphs. We have also compared MMGC and ANTCOL wth state-of-the-art algorthms. Results are presented n Table I. Snce MMGC, the same as ANTCOL, utlzes ANTRLF as the soluton constructon procedure, by properly settng the number of teratons and the colony sze, t wll nearly need the same computatonal tme as ANTCOL does. Nonetheless, n some test nstances MMGC relatvely taes further runnng tme due to applyng Kempe chan local search heurstc. Accordng to table I, MMGC outperforms ANTCOL wth at least one less color n the majorty of the random graphs. However, MMGC results do not approach best nown results on some test nstances partcularly larger ones. Regardng Leghton graphs, for Le_450_5c, both algorthms found the optmal colorng and MMGC outperforms ANTCOL on Le_450_25c. MMGC comparatvely colors Queen 5_5 wth one less color, and n School_nsh, both algorthms reached the best nown result.

10 302 E. Salar and K. Eshgh Graph Best- Known No. Iteratons / No. Ants MMGC No. Success / No. Trals CPU tme (Sec.) ANTCOL G 25, / 70 4/ / 00 5/5 73 G 25, / 70 5/ / 00 4/5 25 G 25, / 70 4/ / 00 4/5 97 G 250, / 20 5/ / 00 5/5 73 G 250, / 20 5/ / 00 3/5 548 G 250, / 20 4/ / 00 3/5 42 G 500, / 20 5/ / 00 5/5 325 G 500, / 20 4/ / 00 3/ G 500, / 20 4/ / 00 4/ Le_450_ 5c / 20 5/ / 00 5/ Le_450_ 25c / 20 5/ / 300 5/5 498 Queen 5_ / 20 3/ / 00 4/5 624 School_ nsh / 20 5/ / 00 5/5 36 TABLE I: MMGC comparson wth ANTCOL and best nown results of the state-of-the-art algorthms for graph colorng. denotes the global-best chromatc numbers found by ANTCOL and MMGC. 'No. Iteratons/ No. Ants' represents the termnaton condton and the colony sze, respectvely. 'No. Success/ No. Trals' demonstrates the number of trals out of 5 runs n whch the reported result s acheved. No. Iteratons / No. Ants No. Success / No. Trals CPU tme (Sec.) VII. CONCLUSION In ths paper, a modfcaton of an exstng ACO algorthm for the graph colorng problem s proposed. Ths modfcaton conforms to Max-Mn ant system structure and explots a local search heurstc to mprove ts performance. It also uses a more promsng evaluaton functon to dstngush between explored colorngs. Expermental results show mprovements over ANTCOL, however, t does not reach the best nown results on some test nstances. Further research attempts are requred to utlze a more effectve local search heurstc and to enhance the ants colorng phase through encouragng them to also produce nfeasble colorngs. It seems that producng only feasble colorngs, durng constructon phase, leads to slender exploraton. We are gong to let ants produce partal nfeasble colorngs durng constructon phase and by applyng an approprate local search heurstc transform them nto feasble ones and mprove them. REFERENCES [] D. Br'elaz, New Methods to Color the Vertces of a Graph, Communcatons of the ACM, 22(979),

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