Abstract
Graph vertex coloring is one of the most studied NP-hard combinatorial optimization problems. Given the hardness of the problem, various heuristic algorithms have been proposed for practical graph coloring, based on local search, population-based approaches and hybrid methods. The research in graph coloring heuristics is very active and improved results have been obtained recently, notably for coloring large and very large graphs. This chapter surveys and analyzes graph coloring heuristics with a focus on the most recent advances.
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References
Aardal, K., van Hoesel, S.P.M., Koster, A.M.C.A., Mannino, C., Sassano, A.: Models and solution techniques for frequency assignment problems. Ann. of Oper. Res. 153(1), 79–129 (2007)
Avanthay, C., Hertz, A., Zufferey, N.: A variable neighborhood search for graph coloring. Eur. J. of Oper. Res. 151(2), 379–388 (2003)
Blöechliger, I., Zufferey, N.: A graph coloring heuristic using partial solutions and a reactive tabu scheme. Comput. & Oper. Res. 35(3), 960–975 (2008)
Blum, C., Roli, A.: Metaheuristics in combinatorial optimization: Overview and conceptual comparison. ACM Comput. Surveys 35(3), 268–308 (2003)
Brélaz, D.: New methods to color the vertices of a graph. Commun. of the ACM 22(4), 251–256 (1979)
Bui, T.N., Nguyen, T.V.H., Patel, C.M., Phan, K.-A.T.: An ant-based algorithm for coloring graphs. Discrete Appl. Math. 156(2), 190–200 (2008)
Chaitin, G.J.: Register allocation and spilling via graph coloring. ACM SIGPLAN Notices 17(6), 98–105 (1982)
Chalupa, D.: Population-based and learning-based metaheuristic algorithms for the graph coloring problem. In: Krasnogor, N., Lanzi, P. (eds.) Proc. of the 13th annual Genet. and Evol. Comput. Conf. (GECCO), Dublin, Ireland, July 12-16, pp. 465–472. ACM Press, N.Y. (2011)
Chams, M., Hertz, A., de Werra, D.: Some experiments with simulated annealing for coloring graphs. Eur. J. of Oper. Res. 32(2), 260–266 (1987)
Chiarandini, M., Stützle, T.: An application of iterated local search to graph coloring. In: Johnson, D., Mehrotra, A., Trick, M. (eds.) Proc. of the Comput. Symp. on Graph Color. and its Gen. (COLOR), Ithaca, N. Y., USA, September 7-8, pp. 112–125 (2002)
Chiarandini, M., Stützle, T.: An Analysis of Heuristics for Vertex Colouring. In: Festa, P. (ed.) SEA 2010. LNCS, vol. 6049, pp. 326–337. Springer, Heidelberg (2010)
Costa, D., Hertz, A., Dubuis, O.: Embedding of a sequential procedure within an evolutionary algorithm for coloring problems in graphs. J. of Heuristics 1(1), 105–128 (1995)
Davis, L.: Order-based genetic algorithms and the graph coloring problem. In: Davis, L. (ed.) Handbook of Genetic Algorithms, pp. 72–90. Van Nostrand Reinhold, N. Y. (1991)
Dorne, R., Hao, J.K.: A New Genetic Local Search Algorithm for Graph Coloring. In: Eiben, A.E., Bäck, T., Schoenauer, M., Schwefel, H.-P. (eds.) PPSN 1998. LNCS, vol. 1498, pp. 745–754. Springer, Heidelberg (1998)
Dzongang, C., Galinier, P., Pierre, S.: A Tabu search heuristic for the routing and wavelength assignment problem in optical networks. IEEE Commun. Lett. 9(5), 426–428 (1998)
Eiben, A., van der Hauw, J., van Hemert, J.: Graph coloring with adaptive evolutionary algorithms. J. of Heuristics 4(1), 24–46 (1998)
Erdős, P., Rényi, A.: On random graphs I. Publ. Math. Debr. 6, 290–297 (1959)
Falkenauer, E.: Genetic Algorithms and Grouping Problems. John Wiley & Sons, Inc., N.Y. (1997)
Fleurent, C., Ferland, J.: Genetic and hybrid algorithms for graph coloring. Ann. of Oper. Res. 63(3), 437–461 (1996)
Galinier, P., Hao, J.K.: Hybrid evolutionary algorithms for graph coloring. J. of Comb. Optim. 3(4), 379–397 (1999)
Galinier, P., Hertz, A.: A survey of local search methods for graph coloring. Comput. & Oper. Res. 33(9), 2547–2562 (2006)
Galinier, P., Hertz, A., Zufferey, N.: An adaptive memory algorithm for the k-coloring problem. Discrete Appl. Math. 156(2), 267–279 (2008)
Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completness. H. Freeman and Co., San Franc. (1979)
Gendron, B., Hertz, A., St-Louis, P.: On edge orienting methods for graph coloring. J. of Comb. Optim. 13(2), 163–178 (2007)
Gendron, B., Hertz, A., St-Louis, P.: On a generalization of the Gallai-Roy-Vitaver theorem to the bandwidth coloring problem. Oper. Res. Lett. 36(1), 345–350 (2008)
Glass, C., Prügel-Bennett, A.: Genetic algorithm for graph coloring: Exploration of Galinier and Hao’s algorithm. J. of Comb. Optim. 7(3), 229–236 (2003)
Gusfield, D.: Partition-distance: A problem and class of perfect graphs arising in clustering. Inf. Process. Lett. 82(3), 159–164 (2002)
Hale, W.K.: Frequency assignment: Theory and applications. IEEE Trans. on Veh. Technol. 68(12), 1497–1514 (1980)
Hamiez, J.-P., Hao, J.-K.: Scatter Search for Graph Coloring. In: Collet, P., Fonlupt, C., Hao, J.-K., Lutton, E., Schoenauer, M. (eds.) EA 2001. LNCS, vol. 2310, pp. 168–179. Springer, Heidelberg (2002)
Hamiez, J.P., Hao, J.K., Glover, F.: A study of tabu search for coloring random 3-colorable graphs around the phase transition. Int. J. of Appl. Metaheuristic Comput. 1(4), 1–24 (2010)
Hao, J.K., Wu, Q.: Improving the extraction and expansion approach for large graph coloring (September 2011); submitted manuscr.
Held, S., Cook, W., Sewell, E.: Safe Lower Bounds for Graph Coloring. In: Günlük, O., Woeginger, G.J. (eds.) IPCO 2011. LNCS, vol. 6655, pp. 261–273. Springer, Heidelberg (2011)
Hertz, A., de Werra, D.: Using tabu search techniques for graph coloring. Comput. 39(4), 345–351 (1987)
Hertz, A., Plumettaz, M., Zufferey, N.: Variable space search for graph coloring. Discrete Appl. Math. 156(13), 2551–2560 (2008)
Hertz, A., Zufferey, N.: A new ant algorithm for graph coloring. In: Proc. of the Workshop on Nat. Inspired Coop. Strateg. for Optim. (NISCO), Granada, Spain, June 29-30, pp. 51–60 (2006)
Jagota, A.: An adaptive, multiple restarts neural network algorithm for graph coloring. Eur. J. of Oper. Res. 93(2), 257–270 (1996)
Janczewski, R.: T-coloring of graphs. In: Contemp. Math., vol. 352, ch. 5, pp. 67–77. Am. Math. Soc., New Providence (2004)
Jensen, T., Toft, B.: Graph Coloring Problems. Wiley-Interscience Ser. in Discrete Math. and Optim. John Wiley & Sons, Inc., N.Y. (1994)
Johnson, D., Aragon, C., McGeoch, L., Schevon, C.: Optimization by simulated annealing: An experimental evaluation; Part II, Graph coloring and number partitioning. Oper. Res. 39(3), 378–406 (1991)
Karp, R.: Reducibility among combinatorial problems, pp. 85–103. Plenum Press, N.Y. (1972)
Korst, J., Aarts, E.: Combinatorial optimization on a boltzmann machine. J. of Parallel and Distrib. Comput. 6(2), 331–357 (1989)
Leighton, F.: A graph coloring algorithm for large scheduling problems. J. of Res. of the Natl. Bureau of Stand. 84(6), 489–506 (1979)
Lü, Z., Hao, J.K.: A memetic algorithm for graph coloring. Eur. J. of Oper. Res. 203(2), 241–250 (2010)
Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. J. of the ACM 41(5), 960–981 (1994)
Malaguti, E., Monaci, M., Toth, P.: A metaheuristic approach for the vertex coloring problem. INFORMS J. on Comput. 20(2), 302–316 (2008)
Malaguti, E., Monaci, M., Toth, P.: An exact approach for the vertex coloring problem. Discrete Optim. 8(2), 174–190 (2011)
Malaguti, E., Toth, P.: An evolutionary approach for bandwidth multicoloring problems. Eur. J. of Oper. Res. 189(3), 638–651 (2008)
Marino, A., Prügel-Bennett, A., Glass, C.: Improving graph colouring with linear programming and genetic algorithms. In: Proc. of the Short Course on Evol. Algorithms in Eng. and Comput. Sci. (EUROGEN), Jyväskylä, Finland, May 30-June 3, pp. 113–118 (1999)
Morgenstern, C.: Distributed coloration neighborhood search. In: Johnson, D., Trick, M. (eds.) Cliques, Coloring, and Satisfiability. DIMACS Ser. in Discrete Math. and Theor. Comput. Sci., vol. 26, pp. 335–357. Am. Math. Soc., New Providence (1996)
Morgenstern, C., Shapiro, H.: Coloration neighborhood structures for general graph coloring. In: Johnson, D. (ed.) Proc. of the 1st Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), San Franc., USA, January 22-24, pp. 226–235. Soc. for Ind. and Appl. Math, Phila. (1990)
Moscato, P.: Memetic algorithms: A short introduction. In: Corne, D., Glover, F., Dorigo, M. (eds.) New Ideas in Optimization, USA, pp. 219–234. McGraw-Hill Educ., N.Y. (1999)
Mumford, C.L.: New Order-Based Crossovers for the Graph Coloring Problem. In: Runarsson, T.P., Beyer, H.-G., Burke, E.K., Merelo-Guervós, J.J., Whitley, L.D., Yao, X. (eds.) PPSN 2006. LNCS, vol. 4193, pp. 880–889. Springer, Heidelberg (2006)
Piwakowski, K.: List coloring of graphs. In: Contemp. Math., vol. 352, ch. 11, pp. 153–162. Am. Math. Soc., New Providence (2004)
Plumettaz, M., Schindl, D., Zufferey, N.: Ant Local Search and its efficient adaptation to graph colouring. J. of the Oper. Res. Soc. 61(5), 819–826 (2010)
Porumbel, D.C., Hao, J.-K., Kuntz, P.: A Study of Evaluation Functions for the Graph K-Coloring Problem. In: Monmarché, N., Talbi, E.-G., Collet, P., Schoenauer, M., Lutton, E. (eds.) EA 2007. LNCS, vol. 4926, pp. 124–135. Springer, Heidelberg (2008)
Porumbel, C., Hao, J., Kuntz, P.: A search space “cartography” for guiding graph coloring heuristics. Comput. & Oper. Res. 37, 769–778 (2010)
Porumbel, D., Hao, J.K., Kuntz, P.: An evolutionary approach with diversity guarantee and well-informed grouping recombination for graph coloring. Comput. & Oper. Res. 37(10), 1822–1832 (2010)
Porumbel, D., Hao, J.K., Kuntz, P.: Spacing memetic algorithms. In: Krasnogor, N., Lanzi, P. (eds.) Proc. of the 13th Annual Genet. and Evol. Comput. Conf. (GECCO), July 12-16, pp. 1061–1068. ACM Press, N.Y. (2011)
Porumbel, D., Hao, J.K., Kuntz, P.: An efficient algorithm for computing the distance between close partitions. Discrete Appl. Math. 159(1), 53–59 (2011)
Radcliffe, N.: The algebra of genetic algorithms. Ann. of Math. and Artif. Intell. 10(4), 339–384 (1994)
Schaerf, A.: A survey of automated timetabling. Artif. Intell. Rev. 13(2), 87–127 (1999)
Shawe-Taylor, J., Žerovnik, J.: Ants and graph coloring. Tech. Rep. 952, Univ. of Ljubljana / Math. Dep., Slovenia (2004), http://www.imfm.si/preprinti/PDF/00952.pdf
Tagawa, K., Kanesige, K., Inoue, K., Haneda, H.: Distance based hybrid genetic algorithm: An application for the graph coloring problem. In: Proc. of the 1999 Congr. on Evol. Comput., pp. 2325–2332 (1999)
Takefuji, Y., Lee, K.: Artificial neural networks for four-coloring map problems and K-colorability problems. IEEE Trans. on Circuits and Syst. 38(3), 326–333 (1991)
Tesman, B.: Set T-colorings of graphs. Congr. Numer. 77, 229–242 (1990)
Titiloye, O., Crispin, A.: Quantum annealing of the graph coloring problem. Discrete Optim. 8(2), 376–384 (2011)
Titiloye, O., Crispin, A.: Graph Coloring with a Distributed Hybrid Quantum Annealing Algorithm. In: O’Shea, J., Nguyen, N.T., Crockett, K., Howlett, R.J., Jain, L.C. (eds.) KES-AMSTA 2011. LNCS, vol. 6682, pp. 553–562. Springer, Heidelberg (2011)
Trick, M.: Appendix: Second DIMACS challenge test problems. DIMACS Ser. in Discrete Math. and Theor. Comput. Sci., vol. 26, pp. 653–657. Am. Math. Soc., New Providence (1996)
Trick, M.A., Yildiz, H.: A Large Neighborhood Search Heuristic for Graph Coloring. In: Van Hentenryck, P., Wolsey, L.A. (eds.) CPAIOR 2007. LNCS, vol. 4510, pp. 346–360. Springer, Heidelberg (2007)
Wu, Q., Hao, J.K.: An adaptive multistart tabu search approach to solve the maximum clique problem (2012), doi:10.1007/s10878-011-9437-8
Wu, Q., Hao, J.K.: Coloring large graphs based on independent set extraction. Comput. & Oper. Res. 39(2), 283–290 (2012)
Wu, Q., Hao, J.K.: An extraction and expansion approach for graph coloring. Asia-Pac. J. of Oper. Res. (in press, 2012)
Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory of Comput. 3, 103–128 (2007)
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Galinier, P., Hamiez, JP., Hao, JK., Porumbel, D. (2013). Recent Advances in Graph Vertex Coloring. In: Zelinka, I., Snášel, V., Abraham, A. (eds) Handbook of Optimization. Intelligent Systems Reference Library, vol 38. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30504-7_20
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