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2018 | OriginalPaper | Buchkapitel

44. The Maximum Clique and Vertex Coloring

verfasst von : Oleksandra Yezerska, Sergiy Butenko

Erschienen in: Handbook of Heuristics

Verlag: Springer International Publishing

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Abstract

In this chapter we review heuristic approaches for two classical and closely related problems of finding a maximum clique and an optimal vertex coloring. Both problems have a wide variety of practical applications, and due to their computational intractability, a significant effort has been focused on developing heuristic methods. This chapter discusses construction heuristics, local search strategies, and metaheuristics designed and/or adapted for the maximum clique and vertex coloring problems.

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Literatur
1.
Zurück zum Zitat Aarts E, Korst J (1988) Simulated annealing and Boltzmann machines. Wiley, Chichester Aarts E, Korst J (1988) Simulated annealing and Boltzmann machines. Wiley, Chichester
2.
Zurück zum Zitat Abello J, Pardalos PM, Resende MGC (1999) On maximum clique problems in very large graphs. In: Abello J, Vitter J (eds) External memory algorithms and visualization. American Mathematical Society, Boston, pp 119–130 Abello J, Pardalos PM, Resende MGC (1999) On maximum clique problems in very large graphs. In: Abello J, Vitter J (eds) External memory algorithms and visualization. American Mathematical Society, Boston, pp 119–130
3.
Zurück zum Zitat Abello J, Butenko S, Pardalos PM, Resende MGC (2001) Finding independent sets in a graph using continuous multivariable polynomial formulations. J Glob Optim 21(2):111–137 Abello J, Butenko S, Pardalos PM, Resende MGC (2001) Finding independent sets in a graph using continuous multivariable polynomial formulations. J Glob Optim 21(2):111–137
4.
Zurück zum Zitat Aggarwal CC, Orlin JB, Tai RP (1997) Optimized crossover for the independent set problem. Oper Res 45(2):226–234 Aggarwal CC, Orlin JB, Tai RP (1997) Optimized crossover for the independent set problem. Oper Res 45(2):226–234
5.
Zurück zum Zitat Andrade DV, Resende MGC, Werneck RF (2012) Fast local search for the maximum independent set problem. J Heuristics 18(4):525–547 Andrade DV, Resende MGC, Werneck RF (2012) Fast local search for the maximum independent set problem. J Heuristics 18(4):525–547
6.
Zurück zum Zitat Arora S, Safra S (1998) Probabilistic checking of proofs: a new characterization of NP. J ACM (JACM) 45(1):70–122 Arora S, Safra S (1998) Probabilistic checking of proofs: a new characterization of NP. J ACM (JACM) 45(1):70–122
7.
Zurück zum Zitat Arora S, Lund C, Motwani R, Sudan M, Szegedy M (1998) Proof verification and the hardness of approximation problems. J ACM (JACM) 45(3):501–555 Arora S, Lund C, Motwani R, Sudan M, Szegedy M (1998) Proof verification and the hardness of approximation problems. J ACM (JACM) 45(3):501–555
8.
Zurück zum Zitat Avanthay C, Hertz A, Zufferey N (2003) A variable neighborhood search for graph coloring. Eur J Oper Res 151(2):379–388 Avanthay C, Hertz A, Zufferey N (2003) A variable neighborhood search for graph coloring. Eur J Oper Res 151(2):379–388
9.
Zurück zum Zitat Back T, Khuri S (1994) An evolutionary heuristic for the maximum independent set problem. In: Proceedings of the first IEEE conference on evolutionary computation, pp 531–535 Back T, Khuri S (1994) An evolutionary heuristic for the maximum independent set problem. In: Proceedings of the first IEEE conference on evolutionary computation, pp 531–535
10.
Zurück zum Zitat Balas E, Niehaus W (1996) Finding large cliques in arbitrary graphs by bipartite matching. DIMACS Ser Discrete Math Theor Comput Sci 26:29–52 Balas E, Niehaus W (1996) Finding large cliques in arbitrary graphs by bipartite matching. DIMACS Ser Discrete Math Theor Comput Sci 26:29–52
11.
Zurück zum Zitat Balas E, Niehaus W (1998) Optimized crossover-based genetic algorithms for the maximum cardinality and maximum weight clique problems. J Heuristics 4(2):107–122 Balas E, Niehaus W (1998) Optimized crossover-based genetic algorithms for the maximum cardinality and maximum weight clique problems. J Heuristics 4(2):107–122
12.
Zurück zum Zitat Balasundaram B, Butenko S (2006) Graph domination, coloring and cliques in telecommunications. In: Handbook of optimization in telecommunications. Springer, Berlin, pp 865–890 Balasundaram B, Butenko S (2006) Graph domination, coloring and cliques in telecommunications. In: Handbook of optimization in telecommunications. Springer, Berlin, pp 865–890
13.
Zurück zum Zitat Battiti R, Mascia F (2010) Reactive and dynamic local search for max-clique: engineering effective building blocks. Comput Oper Res 37(3):534–542 Battiti R, Mascia F (2010) Reactive and dynamic local search for max-clique: engineering effective building blocks. Comput Oper Res 37(3):534–542
14.
Zurück zum Zitat Battiti R, Protasi M (2001) Reactive local search for the maximum clique problem. Algorithmica 29(4):610–637 Battiti R, Protasi M (2001) Reactive local search for the maximum clique problem. Algorithmica 29(4):610–637
15.
Zurück zum Zitat Benlic U, Hao JK (2013) Breakout local search for maximum clique problems. Comput Oper Res 40(1):192–206 Benlic U, Hao JK (2013) Breakout local search for maximum clique problems. Comput Oper Res 40(1):192–206
16.
Zurück zum Zitat Berger MO (1994) k-coloring vertices using a neural network with convergence to valid solutions. In: Proceedings of IEEE international conference on neural networks, vol 7, pp 4514–4517 Berger MO (1994) k-coloring vertices using a neural network with convergence to valid solutions. In: Proceedings of IEEE international conference on neural networks, vol 7, pp 4514–4517
17.
Zurück zum Zitat Blas AD, Jagota A, Hughey R (2002) Energy function-based approaches to graph coloring. IEEE Trans Neural Netw 13(1):81–91 Blas AD, Jagota A, Hughey R (2002) Energy function-based approaches to graph coloring. IEEE Trans Neural Netw 13(1):81–91
18.
Zurück zum Zitat Blöchliger I, Zufferey N (2008) A graph coloring heuristic using partial solutions and a reactive tabu scheme. Comput Oper Res 35(3):960–975 Blöchliger I, Zufferey N (2008) A graph coloring heuristic using partial solutions and a reactive tabu scheme. Comput Oper Res 35(3):960–975
19.
Zurück zum Zitat Blum C, Roli A (2003) Metaheuristics in combinatorial optimization: overview and conceptual comparison. ACM Comput Surv (CSUR) 35(3):268–308 Blum C, Roli A (2003) Metaheuristics in combinatorial optimization: overview and conceptual comparison. ACM Comput Surv (CSUR) 35(3):268–308
20.
Zurück zum Zitat Bollobás B, Thomason A (1985) Random graphs of small order. North-Holland Math Stud 118:47–97 Bollobás B, Thomason A (1985) Random graphs of small order. North-Holland Math Stud 118:47–97
21.
Zurück zum Zitat Bomze IM (1997) Evolution towards the maximum clique. J Glob Optim 10:143–164 Bomze IM (1997) Evolution towards the maximum clique. J Glob Optim 10:143–164
22.
Zurück zum Zitat Bomze IM, Budinich M, Pardalos PM, Pelillo M (1999) The maximum clique problem. In: Handbook of combinatorial optimization. Springer, Boston, pp 1–74 Bomze IM, Budinich M, Pardalos PM, Pelillo M (1999) The maximum clique problem. In: Handbook of combinatorial optimization. Springer, Boston, pp 1–74
23.
Zurück zum Zitat Brélaz D (1979) New methods to color the vertices of a graph. Commun ACM 22(4):251–256 Brélaz D (1979) New methods to color the vertices of a graph. Commun ACM 22(4):251–256
24.
Zurück zum Zitat Brooks RL (1941) On colouring the nodes of a network. In: Mathematical proceedings of the Cambridge philosophical society. Cambridge University Press, vol 37, pp 194–197 Brooks RL (1941) On colouring the nodes of a network. In: Mathematical proceedings of the Cambridge philosophical society. Cambridge University Press, vol 37, pp 194–197
25.
Zurück zum Zitat Brunato M, Battiti R (2011) R-EVO: a reactive evolutionary algorithm for the maximum clique problem. IEEE Trans Evol Comput 15(6):770–782 Brunato M, Battiti R (2011) R-EVO: a reactive evolutionary algorithm for the maximum clique problem. IEEE Trans Evol Comput 15(6):770–782
26.
Zurück zum Zitat Bui TN, Eppley PH (1995) A hybrid genetic algorithm for the maximum clique problem. In: Proceedings of the 6th international conference on genetic algorithms. Morgan Kaufmann Publishers Inc., pp 478–484 Bui TN, Eppley PH (1995) A hybrid genetic algorithm for the maximum clique problem. In: Proceedings of the 6th international conference on genetic algorithms. Morgan Kaufmann Publishers Inc., pp 478–484
27.
Zurück zum Zitat Bui TN, Nguyen TH, Patel CM, Phan KAT (2008) An ant-based algorithm for coloring graphs. Discrete Appl Math 156(2):190–200 Bui TN, Nguyen TH, Patel CM, Phan KAT (2008) An ant-based algorithm for coloring graphs. Discrete Appl Math 156(2):190–200
28.
Zurück zum Zitat Burer S, Monteiro RD, Zhang Y (2002) Maximum stable set formulations and heuristics based on continuous optimization. Math Program 94(1):137–166 Burer S, Monteiro RD, Zhang Y (2002) Maximum stable set formulations and heuristics based on continuous optimization. Math Program 94(1):137–166
29.
Zurück zum Zitat Burke E, Kendall G, Newall J, Hart E, Ross P, Schulenburg S (2003) Hyper-heuristics: an emerging direction in modern search technology. Int Ser Oper Res Manag Sci 57:457–474 Burke E, Kendall G, Newall J, Hart E, Ross P, Schulenburg S (2003) Hyper-heuristics: an emerging direction in modern search technology. Int Ser Oper Res Manag Sci 57:457–474
30.
Zurück zum Zitat Busygin S (2006) A new trust region technique for the maximum weight clique problem. Discrete Appl Math 154(15):2080–2096 Busygin S (2006) A new trust region technique for the maximum weight clique problem. Discrete Appl Math 154(15):2080–2096
31.
Zurück zum Zitat Busygin S, Butenko S, Pardalos PM (2002) A heuristic for the maximum independent set problem based on optimization of a quadratic over a sphere. J Comb Optim 6(3):287–297 Busygin S, Butenko S, Pardalos PM (2002) A heuristic for the maximum independent set problem based on optimization of a quadratic over a sphere. J Comb Optim 6(3):287–297
32.
Zurück zum Zitat Butenko S, Wilhelm WE (2006) Clique-detection models in computational biochemistry and genomics. Eur J Oper Res 173(1):1–17 Butenko S, Wilhelm WE (2006) Clique-detection models in computational biochemistry and genomics. Eur J Oper Res 173(1):1–17
33.
Zurück zum Zitat Butenko S, Pardalos PM, Sergienko IV, Shylo V, Stetsyuk P (2009) Estimating the size of correcting codes using extremal graph problems. In: Pearce C, Hunt E (eds) Optimization: structure and applications. Springer, New York, pp 227–243 Butenko S, Pardalos PM, Sergienko IV, Shylo V, Stetsyuk P (2009) Estimating the size of correcting codes using extremal graph problems. In: Pearce C, Hunt E (eds) Optimization: structure and applications. Springer, New York, pp 227–243
34.
Zurück zum Zitat Butenko S, Yezerska O, Balasundaram B (2013) Variable objective search. J Heuristics 19(4):697–709 Butenko S, Yezerska O, Balasundaram B (2013) Variable objective search. J Heuristics 19(4):697–709
35.
Zurück zum Zitat Caprara A, Kroon L, Monaci M, Peeters M, Toth P (2007) Passenger railway optimization. Handb Oper Res Manag Sci 14:129–187 Caprara A, Kroon L, Monaci M, Peeters M, Toth P (2007) Passenger railway optimization. Handb Oper Res Manag Sci 14:129–187
36.
Zurück zum Zitat Carraghan R, Pardalos PM (1990) An exact algorithm for the maximum clique problem. Oper Res Lett 9(6):375–382 Carraghan R, Pardalos PM (1990) An exact algorithm for the maximum clique problem. Oper Res Lett 9(6):375–382
37.
Zurück zum Zitat Carter R, Park K (1993) How good are genetic algorithms at finding large cliques: an experimental study. Technical report, Computer Science Department, Boston University Carter R, Park K (1993) How good are genetic algorithms at finding large cliques: an experimental study. Technical report, Computer Science Department, Boston University
38.
Zurück zum Zitat Chaitin GJ, Auslander MA, Chandra AK, Cocke J, Hopkins ME, Markstein PW (1981) Register allocation via coloring. Comput Lang 6(1):47–57 Chaitin GJ, Auslander MA, Chandra AK, Cocke J, Hopkins ME, Markstein PW (1981) Register allocation via coloring. Comput Lang 6(1):47–57
39.
Zurück zum Zitat Chams M, Hertz A, de Werra D (1987) Some experiments with simulated annealing for coloring graphs. Eur J Oper Res 32(2):260–266 Chams M, Hertz A, de Werra D (1987) Some experiments with simulated annealing for coloring graphs. Eur J Oper Res 32(2):260–266
40.
Zurück zum Zitat Chiarandini M, Stützle T et al (2002) An application of iterated local search to graph coloring problem. In: Proceedings of the computational symposium on graph coloring and its generalizations, pp 112–125 Chiarandini M, Stützle T et al (2002) An application of iterated local search to graph coloring problem. In: Proceedings of the computational symposium on graph coloring and its generalizations, pp 112–125
41.
Zurück zum Zitat Chiarandini M, Dumitrescu I, Stützle T (2007) Stochastic local search algorithms for the graph colouring problem. In: Handbook of approximation algorithms and metaheuristics. Chapman & Hall/CRC, Boca Raton, pp 63-1 Chiarandini M, Dumitrescu I, Stützle T (2007) Stochastic local search algorithms for the graph colouring problem. In: Handbook of approximation algorithms and metaheuristics. Chapman & Hall/CRC, Boca Raton, pp 63-1
42.
Zurück zum Zitat Chow FC, Hennessy JL (1990) The priority-based coloring approach to register allocation. ACM Trans Program Lang Syst (TOPLAS) 12(4):501–536 Chow FC, Hennessy JL (1990) The priority-based coloring approach to register allocation. ACM Trans Program Lang Syst (TOPLAS) 12(4):501–536
43.
Zurück zum Zitat Costa D, Hertz A (1997) Ants can colour graphs. J Oper Res Soc 48(3):295–305 Costa D, Hertz A (1997) Ants can colour graphs. J Oper Res Soc 48(3):295–305
44.
Zurück zum Zitat Costa D, Hertz A, Dubuis C (1995) Embedding a sequential procedure within an evolutionary algorithm for coloring problems in graphs. J Heuristics 1(1):105–128 Costa D, Hertz A, Dubuis C (1995) Embedding a sequential procedure within an evolutionary algorithm for coloring problems in graphs. J Heuristics 1(1):105–128
45.
Zurück zum Zitat Cottle RW, Pang JS, Stone RE (1992) The linear complementarity problem. SIAM, Philadelphia Cottle RW, Pang JS, Stone RE (1992) The linear complementarity problem. SIAM, Philadelphia
46.
Zurück zum Zitat Culberson JC (1992) Iterated greedy graph coloring and the difficulty landscape. Technical report. TK 92-07, Department of Computing Science, University of Alberta Culberson JC (1992) Iterated greedy graph coloring and the difficulty landscape. Technical report. TK 92-07, Department of Computing Science, University of Alberta
47.
Zurück zum Zitat Culberson JC, Luo F (1996) Exploring the k-colorable landscape with iterated greedy. In: Johnson DS, Trick MA (eds) Cliques, coloring, and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence, pp 245–284 Culberson JC, Luo F (1996) Exploring the k-colorable landscape with iterated greedy. In: Johnson DS, Trick MA (eds) Cliques, coloring, and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence, pp 245–284
48.
Zurück zum Zitat Davis L (1991) Order-based genetic algorithms and the graph coloring problem. In: Handbook of genetic algorithms. Van Nostrand Reinhold, New York, pp 72–90 Davis L (1991) Order-based genetic algorithms and the graph coloring problem. In: Handbook of genetic algorithms. Van Nostrand Reinhold, New York, pp 72–90
51.
Zurück zum Zitat Dorigo M, Maniezzo V, Colorni A (1996) Ant system: optimization by a colony of cooperating agents. IEEE Trans Syst Man Cybern B Cybern 26(1):29–41 Dorigo M, Maniezzo V, Colorni A (1996) Ant system: optimization by a colony of cooperating agents. IEEE Trans Syst Man Cybern B Cybern 26(1):29–41
52.
Zurück zum Zitat Dorne R, Hao JK (1998) A new genetic local search algorithm for graph coloring. In: Parallel problem solving from nature. Springer, Berlin/Heidelberg, pp 745–754 Dorne R, Hao JK (1998) A new genetic local search algorithm for graph coloring. In: Parallel problem solving from nature. Springer, Berlin/Heidelberg, pp 745–754
53.
Zurück zum Zitat Dowsland KA, Thompson JM (2005) Ant colony optimization for the examination scheduling problem. J Oper Res Soc 56(4):426–438 Dowsland KA, Thompson JM (2005) Ant colony optimization for the examination scheduling problem. J Oper Res Soc 56(4):426–438
54.
Zurück zum Zitat Dowsland KA, Thompson JM (2008) An improved ant colony optimisation heuristic for graph colouring. Discrete Appl Math 156(3):313–324 Dowsland KA, Thompson JM (2008) An improved ant colony optimisation heuristic for graph colouring. Discrete Appl Math 156(3):313–324
55.
Zurück zum Zitat Dukanovic I, Rendl F (2007) Semidefinite programming relaxations for graph coloring and maximal clique problems. Math Program 109(2–3):345–365 Dukanovic I, Rendl F (2007) Semidefinite programming relaxations for graph coloring and maximal clique problems. Math Program 109(2–3):345–365
56.
Zurück zum Zitat Dukanovic I, Rendl F (2008) A semidefinite programming-based heuristic for graph coloring. Discrete Appl Math 156(2):180–189 Dukanovic I, Rendl F (2008) A semidefinite programming-based heuristic for graph coloring. Discrete Appl Math 156(2):180–189
57.
Zurück zum Zitat Eiben ÁE, Van Der Hauw JK, van Hemert JI (1998) Graph coloring with adaptive evolutionary algorithms. J Heuristics 4(1):25–46 Eiben ÁE, Van Der Hauw JK, van Hemert JI (1998) Graph coloring with adaptive evolutionary algorithms. J Heuristics 4(1):25–46
58.
Zurück zum Zitat Erdös P (1970) On the graph theorem of Turán. Mat Lapok 21(249–251):10 Erdös P (1970) On the graph theorem of Turán. Mat Lapok 21(249–251):10
59.
Zurück zum Zitat Feige U, Kilian J (1996) Zero knowledge and the chromatic number. In: Proceedings of eleventh annual IEEE conference on computational complexity, pp 278–287 Feige U, Kilian J (1996) Zero knowledge and the chromatic number. In: Proceedings of eleventh annual IEEE conference on computational complexity, pp 278–287
60.
Zurück zum Zitat Feige U, Kilian J (1998) Zero knowledge and the chromatic number. J Comput Syst Sci 57:187–199 Feige U, Kilian J (1998) Zero knowledge and the chromatic number. J Comput Syst Sci 57:187–199
61.
Zurück zum Zitat Fenet S, Solnon C (2003) Searching for maximum cliques with ant colony optimization. In: Applications of evolutionary computing. Springer, Berlin, pp 236–245 Fenet S, Solnon C (2003) Searching for maximum cliques with ant colony optimization. In: Applications of evolutionary computing. Springer, Berlin, pp 236–245
62.
Zurück zum Zitat Feo TA, Resende MGC (1989) A probabilistic heuristic for a computationally difficult set covering problem. Oper Res Lett 8:67–71 Feo TA, Resende MGC (1989) A probabilistic heuristic for a computationally difficult set covering problem. Oper Res Lett 8:67–71
63.
Zurück zum Zitat Feo TA, Resende MGC (1995) Greedy randomized adaptive search procedures. J Glob Optim 6:109–133 Feo TA, Resende MGC (1995) Greedy randomized adaptive search procedures. J Glob Optim 6:109–133
64.
Zurück zum Zitat Feo TA, Resende MGC, Smith SH (1994) A greedy randomized adaptive search procedure for maximum independent set. Oper Res 42(5):860–878 Feo TA, Resende MGC, Smith SH (1994) A greedy randomized adaptive search procedure for maximum independent set. Oper Res 42(5):860–878
65.
Zurück zum Zitat Ferland J, Fleurent C (1996) Object-oriented implementation of heuristic search methods for graph coloring, maximum clique and satisfiability. In: Johnson DS, Trick MA (eds) Cliques, coloring, and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence, pp 619–652 Ferland J, Fleurent C (1996) Object-oriented implementation of heuristic search methods for graph coloring, maximum clique and satisfiability. In: Johnson DS, Trick MA (eds) Cliques, coloring, and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence, pp 619–652
66.
Zurück zum Zitat Fleurent C, Ferland JA (1996) Genetic and hybrid algorithms for graph coloring. Ann Oper Res 63(3):437–461 Fleurent C, Ferland JA (1996) Genetic and hybrid algorithms for graph coloring. Ann Oper Res 63(3):437–461
67.
Zurück zum Zitat Foster JA, Soule T (1995) Using genetic algorithms to find maximum cliques. Technical report. LAL95-12, Department of Computer Science, University of Idaho Foster JA, Soule T (1995) Using genetic algorithms to find maximum cliques. Technical report. LAL95-12, Department of Computer Science, University of Idaho
68.
Zurück zum Zitat Friden C, Hertz A, de Werra D (1989) STABULUS: a technique for finding stable sets in large graphs with tabu search. Computing 42:35–44 Friden C, Hertz A, de Werra D (1989) STABULUS: a technique for finding stable sets in large graphs with tabu search. Computing 42:35–44
69.
Zurück zum Zitat Funabiki N, Higashino T (2000) A minimal-state processing search algorithm for graph coloring problems. IEICE Trans Fundam Electron Commun Comput Sci 83(7):1420–1430 Funabiki N, Higashino T (2000) A minimal-state processing search algorithm for graph coloring problems. IEICE Trans Fundam Electron Commun Comput Sci 83(7):1420–1430
70.
Zurück zum Zitat Galinier P, Hao JK (1999) Hybrid evolutionary algorithms for graph coloring. J Comb Optim 3(4):379–397 Galinier P, Hao JK (1999) Hybrid evolutionary algorithms for graph coloring. J Comb Optim 3(4):379–397
71.
Zurück zum Zitat Galinier P, Hertz A (2006) A survey of local search methods for graph coloring. Comput Oper Res 33(9):2547–2562 Galinier P, Hertz A (2006) A survey of local search methods for graph coloring. Comput Oper Res 33(9):2547–2562
72.
Zurück zum Zitat Galinier P, Hertz A, Zufferey N (2008) An adaptive memory algorithm for the k-coloring problem. Discrete Appl Math 156(2):267–279 Galinier P, Hertz A, Zufferey N (2008) An adaptive memory algorithm for the k-coloring problem. Discrete Appl Math 156(2):267–279
73.
Zurück zum Zitat Galinier P, Hamiez JP, Hao JK, Porumbel D (2013) Recent advances in graph vertex coloring. In: Handbook of optimization. Springer, Berlin/Heidelberg, pp 505–528 Galinier P, Hamiez JP, Hao JK, Porumbel D (2013) Recent advances in graph vertex coloring. In: Handbook of optimization. Springer, Berlin/Heidelberg, pp 505–528
74.
Zurück zum Zitat Gamst A (1986) Some lower bounds for a class of frequency assignment problems. IEEE Trans Veh Technol 35(1):8–14 Gamst A (1986) Some lower bounds for a class of frequency assignment problems. IEEE Trans Veh Technol 35(1):8–14
75.
Zurück zum Zitat Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. W.H. Freeman and Company, New York Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. W.H. Freeman and Company, New York
76.
Zurück zum Zitat Garey MR, Johnson DS, So H (1976) An application of graph coloring to printed circuit testing. IEEE Trans Circuits Syst 23(10):591–599 Garey MR, Johnson DS, So H (1976) An application of graph coloring to printed circuit testing. IEEE Trans Circuits Syst 23(10):591–599
77.
Zurück zum Zitat Gassen DW, Carothers JD (1993) Graph color minimization using neural networks. In: Proceedings of IEEE international joint conference on neural networks, vol 2, pp 1541–1544 Gassen DW, Carothers JD (1993) Graph color minimization using neural networks. In: Proceedings of IEEE international joint conference on neural networks, vol 2, pp 1541–1544
78.
Zurück zum Zitat Gendreau M, Soriano P, Salvail L (1993) Solving the maximum clique problem using a tabu search approach. Ann Oper Res 41(4):385–403 Gendreau M, Soriano P, Salvail L (1993) Solving the maximum clique problem using a tabu search approach. Ann Oper Res 41(4):385–403
79.
Zurück zum Zitat Gibbons LE, Hearn DW, Pardalos PM (1996) A continuous based heuristic for the maximum clique problem. In: Johnson DS, Trick MA (eds) Cliques, coloring, and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence, pp 103–124 Gibbons LE, Hearn DW, Pardalos PM (1996) A continuous based heuristic for the maximum clique problem. In: Johnson DS, Trick MA (eds) Cliques, coloring, and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence, pp 103–124
80.
Zurück zum Zitat Glass CA, Prügel-Bennett A (2003) Genetic algorithm for graph coloring: exploration of Galinier and Hao’s algorithm. J Comb Optim 7(3):229–236 Glass CA, Prügel-Bennett A (2003) Genetic algorithm for graph coloring: exploration of Galinier and Hao’s algorithm. J Comb Optim 7(3):229–236
81.
Zurück zum Zitat Glover F (1989) Tabu search. Part I. ORSA J Comput 1(3):190–206 Glover F (1989) Tabu search. Part I. ORSA J Comput 1(3):190–206
82.
Zurück zum Zitat Goldberg DE (1989) Genetic algorithms in search, optimization, and machine learning. Addison-Wesley, Reading Goldberg DE (1989) Genetic algorithms in search, optimization, and machine learning. Addison-Wesley, Reading
83.
Zurück zum Zitat Goldberg MK, Rivenburgh RD (1996) Constructing cliques using restricted backtracking. In: Johnson DS, Trick MA (eds) Cliques, coloring, and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence, pp 285–307 Goldberg MK, Rivenburgh RD (1996) Constructing cliques using restricted backtracking. In: Johnson DS, Trick MA (eds) Cliques, coloring, and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence, pp 285–307
84.
Zurück zum Zitat Govorčin J, Gvozdenović N, Povh J (2013) New heuristics for the vertex coloring problem based on semidefinite programming. Cent Eur J Oper Res 21(1):13–25 Govorčin J, Gvozdenović N, Povh J (2013) New heuristics for the vertex coloring problem based on semidefinite programming. Cent Eur J Oper Res 21(1):13–25
85.
Zurück zum Zitat Grable DA, Panconesi A (1998) Fast distributed algorithms for Brooks-Vizing colourings. In: Proceedings of the ninth annual ACM-SIAM symposium on discrete algorithms. Society for Industrial and Applied Mathematics, pp 473–480 Grable DA, Panconesi A (1998) Fast distributed algorithms for Brooks-Vizing colourings. In: Proceedings of the ninth annual ACM-SIAM symposium on discrete algorithms. Society for Industrial and Applied Mathematics, pp 473–480
86.
Zurück zum Zitat Grossman T (1996) Applying the inn model to the maximum clique problem. In: Johnson DS, Trick MA (eds) Cliques, coloring, and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence, pp 125–146 Grossman T (1996) Applying the inn model to the maximum clique problem. In: Johnson DS, Trick MA (eds) Cliques, coloring, and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence, pp 125–146
87.
Zurück zum Zitat Grosso A, Locatelli M, Della Croce F (2004) Combining swaps and node weights in an adaptive greedy approach for the maximum clique problem. J Heuristics 10(2):135–152 Grosso A, Locatelli M, Della Croce F (2004) Combining swaps and node weights in an adaptive greedy approach for the maximum clique problem. J Heuristics 10(2):135–152
88.
Zurück zum Zitat Grosso A, Locatelli M, Pullan W (2008) Simple ingredients leading to very efficient heuristics for the maximum clique problem. J Heuristics 14(6):587–612 Grosso A, Locatelli M, Pullan W (2008) Simple ingredients leading to very efficient heuristics for the maximum clique problem. J Heuristics 14(6):587–612
89.
Zurück zum Zitat Gruzdeva TV (2013) On a continuous approach for the maximum weighted clique problem. J Glob Optim 56(3):971–981 Gruzdeva TV (2013) On a continuous approach for the maximum weighted clique problem. J Glob Optim 56(3):971–981
90.
Zurück zum Zitat Guturu P, Dantu R (2008) An impatient evolutionary algorithm with probabilistic tabu search for unified solution of some np-hard problems in graph and set theory via clique finding. IEEE Trans Syst Man Cybern B Cybern 38(3):645–666 Guturu P, Dantu R (2008) An impatient evolutionary algorithm with probabilistic tabu search for unified solution of some np-hard problems in graph and set theory via clique finding. IEEE Trans Syst Man Cybern B Cybern 38(3):645–666
91.
Zurück zum Zitat Hajnal P, Szemerédi E (1990) Brooks coloring in parallel. SIAM J Discret Math 3(1):74–80 Hajnal P, Szemerédi E (1990) Brooks coloring in parallel. SIAM J Discret Math 3(1):74–80
92.
Zurück zum Zitat Hamiez JP, Hao JK (2002) Scatter search for graph coloring. In: Artificial evolution. Springer, Berlin/Heidelberg pp 168–179 Hamiez JP, Hao JK (2002) Scatter search for graph coloring. In: Artificial evolution. Springer, Berlin/Heidelberg pp 168–179
93.
Zurück zum Zitat Hansen P, Jaumard B (1990) Algorithms for the maximum satisfiability problem. Computing 44(4):279–303 Hansen P, Jaumard B (1990) Algorithms for the maximum satisfiability problem. Computing 44(4):279–303
94.
Zurück zum Zitat Hansen P, Mladenović N (1999) An introduction to variable neighborhood search. In: Voss S et al (eds) Meta-heuristics: advances and trends in local search paradigms for optimization. Springer, Boston, pp 433–458 Hansen P, Mladenović N (1999) An introduction to variable neighborhood search. In: Voss S et al (eds) Meta-heuristics: advances and trends in local search paradigms for optimization. Springer, Boston, pp 433–458
95.
Zurück zum Zitat Hansen P, Mladenović N, Urošević D (2004) Variable neighborhood search for the maximum clique. Discret Appl Math 145(1):117–125 Hansen P, Mladenović N, Urošević D (2004) Variable neighborhood search for the maximum clique. Discret Appl Math 145(1):117–125
96.
Zurück zum Zitat Hao JK, Wu Q (2012) Improving the extraction and expansion method for large graph coloring. Discret Appl Math 160(16):2397–2407 Hao JK, Wu Q (2012) Improving the extraction and expansion method for large graph coloring. Discret Appl Math 160(16):2397–2407
97.
Zurück zum Zitat Håstad J (1999) Clique is hard to approximate within n1−𝜖. Acta Math 182:105–142 Håstad J (1999) Clique is hard to approximate within n1−𝜖. Acta Math 182:105–142
98.
Zurück zum Zitat Held S, Cook W, Sewell EC (2012) Maximum-weight stable sets and safe lower bounds for graph coloring. Math Program Comput 4(4):363–381 Held S, Cook W, Sewell EC (2012) Maximum-weight stable sets and safe lower bounds for graph coloring. Math Program Comput 4(4):363–381
99.
Zurück zum Zitat Hertz A, de Werra D (1987) Using tabu search techniques for graph coloring. Computing 39(4):345–351 Hertz A, de Werra D (1987) Using tabu search techniques for graph coloring. Computing 39(4):345–351
100.
Zurück zum Zitat Hertz A, Zufferey N (2006) A new ant algorithm for graph coloring. In: Workshop on nature inspired cooperative strategies for optimization, NICSO, pp 51–60 Hertz A, Zufferey N (2006) A new ant algorithm for graph coloring. In: Workshop on nature inspired cooperative strategies for optimization, NICSO, pp 51–60
101.
Zurück zum Zitat Hertz A, Plumettaz M, Zufferey N (2008) Variable space search for graph coloring. Discret Appl Math 156(13):2551–2560 Hertz A, Plumettaz M, Zufferey N (2008) Variable space search for graph coloring. Discret Appl Math 156(13):2551–2560
102.
Zurück zum Zitat Hifi M (1997) A genetic algorithm-based heuristic for solving the weighted maximum independent set and some equivalent problems. J Oper Res Soc 48(6):612–622 Hifi M (1997) A genetic algorithm-based heuristic for solving the weighted maximum independent set and some equivalent problems. J Oper Res Soc 48(6):612–622
103.
Zurück zum Zitat Holland JH (1992) Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence. MIT Press, Cambridge Holland JH (1992) Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence. MIT Press, Cambridge
104.
Zurück zum Zitat Homer S, Peinado M (1996) Experiments with polynomial-time clique approximation algorithms on very large graphs. DIMACS Ser Discret Math Theor Comput Sci 26: 147–168 Homer S, Peinado M (1996) Experiments with polynomial-time clique approximation algorithms on very large graphs. DIMACS Ser Discret Math Theor Comput Sci 26: 147–168
105.
Zurück zum Zitat Horst R, Thoai NV (1999) Dc programming: overview. J Optim Theory Appl 103(1):1–43 Horst R, Thoai NV (1999) Dc programming: overview. J Optim Theory Appl 103(1):1–43
106.
Zurück zum Zitat Jagota A (1992) Efficiently approximating MAX-CLIQUE in a Hopfield-style network. In: International joint conference on neural networks, vol 2, pp 248–253 Jagota A (1992) Efficiently approximating MAX-CLIQUE in a Hopfield-style network. In: International joint conference on neural networks, vol 2, pp 248–253
107.
Zurück zum Zitat Jagota A (1995) Approximating maximum clique with a Hopfield network. IEEE Trans Neural Netw 6(3):724–735 Jagota A (1995) Approximating maximum clique with a Hopfield network. IEEE Trans Neural Netw 6(3):724–735
108.
Zurück zum Zitat Jagota A (1996) An adaptive, multiple restarts neural network algorithm for graph coloring. Eur J Oper Res 93(2):257–270 Jagota A (1996) An adaptive, multiple restarts neural network algorithm for graph coloring. Eur J Oper Res 93(2):257–270
109.
Zurück zum Zitat Jagota A, Sanchis LA (2001) Adaptive, restart, randomized greedy heuristics for maximum clique. J Heuristics 7(6):565–585 Jagota A, Sanchis LA (2001) Adaptive, restart, randomized greedy heuristics for maximum clique. J Heuristics 7(6):565–585
110.
Zurück zum Zitat Jagota A, Sanchis L, Ganesan R (1996) Approximately solving maximum clique using neural networks and related heuristics. DIMACS Ser Discret Math Theor Comput Sci 26: 169–204 Jagota A, Sanchis L, Ganesan R (1996) Approximately solving maximum clique using neural networks and related heuristics. DIMACS Ser Discret Math Theor Comput Sci 26: 169–204
111.
Zurück zum Zitat Jerrum M (1992) Large cliques elude the metropolis process. Random Struct Algorithm 3(4):347–359 Jerrum M (1992) Large cliques elude the metropolis process. Random Struct Algorithm 3(4):347–359
112.
Zurück zum Zitat Jin Y, Hao JK (2015) General swap-based multiple neighborhood tabu search for the maximum independent set problem. Eng Appl Artif Intell 37:20–33 Jin Y, Hao JK (2015) General swap-based multiple neighborhood tabu search for the maximum independent set problem. Eng Appl Artif Intell 37:20–33
113.
Zurück zum Zitat Johnson DS, Trick MA (eds) (1996) Cliques, coloring, and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence Johnson DS, Trick MA (eds) (1996) Cliques, coloring, and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence
114.
Zurück zum Zitat Johnson DS, Aragon CR, McGeoch LA, Schevon C (1991) Optimization by simulated annealing: an experimental evaluation; part II, graph coloring and number partitioning. Oper Res 39(3):378–406 Johnson DS, Aragon CR, McGeoch LA, Schevon C (1991) Optimization by simulated annealing: an experimental evaluation; part II, graph coloring and number partitioning. Oper Res 39(3):378–406
116.
Zurück zum Zitat Karchmer M, Naor J (1988) A fast parallel algorithm to color a graph with Δ colors. J Algorithm 9(1):83–91 Karchmer M, Naor J (1988) A fast parallel algorithm to color a graph with Δ colors. J Algorithm 9(1):83–91
117.
Zurück zum Zitat Karger D, Motwani R, Sudan M (1998) Approximate graph coloring by semidefinite programming. J ACM (JACM) 45(2):246–265 Karger D, Motwani R, Sudan M (1998) Approximate graph coloring by semidefinite programming. J ACM (JACM) 45(2):246–265
118.
Zurück zum Zitat Karloff HJ (1989) An NC algorithm for Brooks’ theorem. Theor Comput Sci 68(1):89–103 Karloff HJ (1989) An NC algorithm for Brooks’ theorem. Theor Comput Sci 68(1):89–103
119.
Zurück zum Zitat Karp RM (1972) Reducibility among combinatorial problems. In: Miller RE, Thatcher JW (eds) Complexity of computer computations. Plenum, New York, pp 85–103 Karp RM (1972) Reducibility among combinatorial problems. In: Miller RE, Thatcher JW (eds) Complexity of computer computations. Plenum, New York, pp 85–103
120.
Zurück zum Zitat Katayama K, Hamamoto A, Narihisa H (2005) An effective local search for the maximum clique problem. Inf Process Lett 95(5):503–511 Katayama K, Hamamoto A, Narihisa H (2005) An effective local search for the maximum clique problem. Inf Process Lett 95(5):503–511
121.
Zurück zum Zitat Kirkpatrick S, Vecchi M et al (1983) Optimization by simulated annealing. Science 220(4598):671–680 Kirkpatrick S, Vecchi M et al (1983) Optimization by simulated annealing. Science 220(4598):671–680
123.
Zurück zum Zitat Kopf R, Ruhe G (1987) A computational study of the weighted independent set problem for general graphs. Found Control Eng 12(4):167–180 Kopf R, Ruhe G (1987) A computational study of the weighted independent set problem for general graphs. Found Control Eng 12(4):167–180
124.
Zurück zum Zitat Laguna M, Martí R (2001) A grasp for coloring sparse graphs. Computat Optim Appl 19(2):165–178 Laguna M, Martí R (2001) A grasp for coloring sparse graphs. Computat Optim Appl 19(2):165–178
125.
Zurück zum Zitat Leighton FT (1979) A graph coloring algorithm for large scheduling problems. J Res Natl Bur Stand 84(6):489–506 Leighton FT (1979) A graph coloring algorithm for large scheduling problems. J Res Natl Bur Stand 84(6):489–506
126.
Zurück zum Zitat Lemke CE (1965) Bimatrix equilibrium points and mathematical programming. Manag Sci 11(7):681–689 Lemke CE (1965) Bimatrix equilibrium points and mathematical programming. Manag Sci 11(7):681–689
128.
Zurück zum Zitat Lovász L (1975) Three short proofs in graph theory. J Comb Theory Ser B 19(3):269–271 Lovász L (1975) Three short proofs in graph theory. J Comb Theory Ser B 19(3):269–271
129.
Zurück zum Zitat Lovász L (1979) On the shannon capacity of a graph. IEEE Trans Inf Theory 25(1):1–7 Lovász L (1979) On the shannon capacity of a graph. IEEE Trans Inf Theory 25(1):1–7
130.
Zurück zum Zitat Lovász L, Plummer MD (2009) Matching theory. American Mathematical Society, Providence Lovász L, Plummer MD (2009) Matching theory. American Mathematical Society, Providence
131.
Zurück zum Zitat Lü Z, Hao JK (2010) A memetic algorithm for graph coloring. Eur J Oper Res 203(1): 241–250 Lü Z, Hao JK (2010) A memetic algorithm for graph coloring. Eur J Oper Res 203(1): 241–250
132.
Zurück zum Zitat Malaguti E, Toth P (2010) A survey on vertex coloring problems. Int Trans Oper Res 17(1): 1–34 Malaguti E, Toth P (2010) A survey on vertex coloring problems. Int Trans Oper Res 17(1): 1–34
133.
Zurück zum Zitat Malaguti E, Monaci M, Toth P (2008) A metaheuristic approach for the vertex coloring problem. INFORMS J Comput 20(2):302–316 Malaguti E, Monaci M, Toth P (2008) A metaheuristic approach for the vertex coloring problem. INFORMS J Comput 20(2):302–316
134.
Zurück zum Zitat Malaguti E, Monaci M, Toth P (2011) An exact approach for the vertex coloring problem. Discret Optim 8(2):174–190 Malaguti E, Monaci M, Toth P (2011) An exact approach for the vertex coloring problem. Discret Optim 8(2):174–190
135.
Zurück zum Zitat Mannino C, Sassano A (1996) Edge projection and the maximum cardinality stable set problem. DIMACS Ser Discret Math Theor Comput Sci 26:205–219 Mannino C, Sassano A (1996) Edge projection and the maximum cardinality stable set problem. DIMACS Ser Discret Math Theor Comput Sci 26:205–219
136.
Zurück zum Zitat Marchiori E (1998) A simple heuristic based genetic algorithm for the maximum clique problem. In: Proceedings of the ACM symposium on applied computing, vol 27, pp 366–373 Marchiori E (1998) A simple heuristic based genetic algorithm for the maximum clique problem. In: Proceedings of the ACM symposium on applied computing, vol 27, pp 366–373
137.
Zurück zum Zitat Marchiori E (2002) Genetic, iterated and multistart local search for the maximum clique problem. In: Applications of evolutionary computing. Springer, Berlin/Heidelberg, pp 112–121 Marchiori E (2002) Genetic, iterated and multistart local search for the maximum clique problem. In: Applications of evolutionary computing. Springer, Berlin/Heidelberg, pp 112–121
138.
Zurück zum Zitat Massaro A, Pelillo M, Bomze IM (2002) A complementary pivoting approach to the maximum weight clique problem. SIAM J Optim 12(4):928–948 Massaro A, Pelillo M, Bomze IM (2002) A complementary pivoting approach to the maximum weight clique problem. SIAM J Optim 12(4):928–948
139.
Zurück zum Zitat Matula DW, Marble G, Isaacson JD (1972) Graph coloring algorithms. In: Read R (ed) Graph theory and computing. Academic, New York, pp 109–122 Matula DW, Marble G, Isaacson JD (1972) Graph coloring algorithms. In: Read R (ed) Graph theory and computing. Academic, New York, pp 109–122
140.
Zurück zum Zitat Mehta NK (1981) The application of a graph coloring method to an examination scheduling problem. Interfaces 11(5):57–65 Mehta NK (1981) The application of a graph coloring method to an examination scheduling problem. Interfaces 11(5):57–65
141.
Zurück zum Zitat Mladenović N, Hansen P (1997) Variable neighborhood search. Comput Oper Res 24(11):1097–1100 Mladenović N, Hansen P (1997) Variable neighborhood search. Comput Oper Res 24(11):1097–1100
142.
Zurück zum Zitat Morgenstern C (1996) Distributed coloration neighborhood search. Discret Math Theor Comput Sci 26:335–358 Morgenstern C (1996) Distributed coloration neighborhood search. Discret Math Theor Comput Sci 26:335–358
143.
Zurück zum Zitat Morgenstern CA, Shapiro HD (1986) Chromatic number approximation using simulated annealing. Technical report, Department of Computer Science, University of New Mexico Morgenstern CA, Shapiro HD (1986) Chromatic number approximation using simulated annealing. Technical report, Department of Computer Science, University of New Mexico
144.
Zurück zum Zitat Motzkin TS, Straus EG (1965) Maxima for graphs and a new proof of a theorem of turán. Canad J Math 17(4):533–540 Motzkin TS, Straus EG (1965) Maxima for graphs and a new proof of a theorem of turán. Canad J Math 17(4):533–540
145.
Zurück zum Zitat Mumford CL (2006) New order-based crossovers for the graph coloring problem. In: Parallel problem solving from nature. Springer, Berlin, pp 880–889 Mumford CL (2006) New order-based crossovers for the graph coloring problem. In: Parallel problem solving from nature. Springer, Berlin, pp 880–889
146.
Zurück zum Zitat Murthy AS, Parthasarathy G, Sastry V (1994) Clique finding – a genetic approach. In: Proceedings of the first IEEE conference on evolutionary computation, pp 18–21 Murthy AS, Parthasarathy G, Sastry V (1994) Clique finding – a genetic approach. In: Proceedings of the first IEEE conference on evolutionary computation, pp 18–21
147.
Zurück zum Zitat Ouyang Q, Kaplan PD, Liu S, Libchaber A (1997) Dna solution of the maximal clique problem. Science 278(5337):446–449 Ouyang Q, Kaplan PD, Liu S, Libchaber A (1997) Dna solution of the maximal clique problem. Science 278(5337):446–449
148.
Zurück zum Zitat Paquete L, Stützle T (2002) An experimental investigation of iterated local search for coloring graphs. In: Applications of evolutionary computing. Springer, Berlin/Heidelberg, pp 122–131 Paquete L, Stützle T (2002) An experimental investigation of iterated local search for coloring graphs. In: Applications of evolutionary computing. Springer, Berlin/Heidelberg, pp 122–131
149.
Zurück zum Zitat Pardalos PM, Phillips A (1990) A global optimization approach for solving the maximum clique problem. Int J Comput Math 33(3–4):209–216 Pardalos PM, Phillips A (1990) A global optimization approach for solving the maximum clique problem. Int J Comput Math 33(3–4):209–216
150.
Zurück zum Zitat Pardalos PM, Xue J (1994) The maximum clique problem. J Glob Optim 4(3):301–328 Pardalos PM, Xue J (1994) The maximum clique problem. J Glob Optim 4(3):301–328
151.
Zurück zum Zitat Pardalos PM, Mavridou T, Xue J (1999) The graph coloring problem: a bibliographic survey. In: Handbook of combinatorial optimization. Springer, Boston, pp 1077–1141 Pardalos PM, Mavridou T, Xue J (1999) The graph coloring problem: a bibliographic survey. In: Handbook of combinatorial optimization. Springer, Boston, pp 1077–1141
152.
Zurück zum Zitat Park K, Carter B (1995) On the effectiveness of genetic search in combinatorial optimization. In: Proceedings of the ACM symposium on applied computing, pp 329–336 Park K, Carter B (1995) On the effectiveness of genetic search in combinatorial optimization. In: Proceedings of the ACM symposium on applied computing, pp 329–336
153.
Zurück zum Zitat Pattillo J, Butenko S (2011) Clique, independent set, and graph coloring. In: Encyclopedia of operations research and management science. Wiley, Hoboken, pp 3150–3163 Pattillo J, Butenko S (2011) Clique, independent set, and graph coloring. In: Encyclopedia of operations research and management science. Wiley, Hoboken, pp 3150–3163
154.
Zurück zum Zitat Philipsen W, Stok L (1991) Graph coloring using neural networks. In: IEEE international symposium on circuits and systems, pp 1597–1600 Philipsen W, Stok L (1991) Graph coloring using neural networks. In: IEEE international symposium on circuits and systems, pp 1597–1600
155.
Zurück zum Zitat Plumettaz M, Schindl D, Zufferey N (2010) Ant local search and its efficient adaptation to graph colouring. J Oper Res Soc 61(5):819–826CrossRef Plumettaz M, Schindl D, Zufferey N (2010) Ant local search and its efficient adaptation to graph colouring. J Oper Res Soc 61(5):819–826CrossRef
156.
Zurück zum Zitat Porumbel DC, Hao JK, Kuntz P (2010) An evolutionary approach with diversity guarantee and well-informed grouping recombination for graph coloring. Comput Oper Res 37(10):1822–1832CrossRef Porumbel DC, Hao JK, Kuntz P (2010) An evolutionary approach with diversity guarantee and well-informed grouping recombination for graph coloring. Comput Oper Res 37(10):1822–1832CrossRef
157.
Zurück zum Zitat Porumbel DC, Hao JK, Kuntz P (2010) A search space “cartography” for guiding graph coloring heuristics. Comput Oper Res 37(4):769–778MathSciNetCrossRef Porumbel DC, Hao JK, Kuntz P (2010) A search space “cartography” for guiding graph coloring heuristics. Comput Oper Res 37(4):769–778MathSciNetCrossRef
158.
159.
Zurück zum Zitat Pullan W, Hoos HH (2006) Dynamic local search for the maximum clique problem. J Artif Intell Res 25:159–185CrossRef Pullan W, Hoos HH (2006) Dynamic local search for the maximum clique problem. J Artif Intell Res 25:159–185CrossRef
160.
Zurück zum Zitat Pullan W, Mascia F, Brunato M (2011) Cooperating local search for the maximum clique problem. J Heuristics 17(2):181–199CrossRef Pullan W, Mascia F, Brunato M (2011) Cooperating local search for the maximum clique problem. J Heuristics 17(2):181–199CrossRef
161.
Zurück zum Zitat Resende MGC, Feo TA, Smith SH (1998) Algorithm 787: fortran subroutines for approximate solution of maximum independent set problems using grasp. ACM Trans Math Softw (TOMS) 24(4):386–394CrossRef Resende MGC, Feo TA, Smith SH (1998) Algorithm 787: fortran subroutines for approximate solution of maximum independent set problems using grasp. ACM Trans Math Softw (TOMS) 24(4):386–394CrossRef
162.
Zurück zum Zitat Singh A, Gupta AK (2006) A hybrid heuristic for the maximum clique problem. J Heuristics 12(1–2):5–22CrossRef Singh A, Gupta AK (2006) A hybrid heuristic for the maximum clique problem. J Heuristics 12(1–2):5–22CrossRef
163.
165.
Zurück zum Zitat Solnon C, Fenet S (2006) A study of ACO capabilities for solving the maximum clique problem. J Heuristics 12(3):155–180CrossRef Solnon C, Fenet S (2006) A study of ACO capabilities for solving the maximum clique problem. J Heuristics 12(3):155–180CrossRef
166.
Zurück zum Zitat Soriano P, Gendreau M (1996) Diversification strategies in tabu search algorithms for the maximum clique problem. Ann Oper Res 63(2):189–207CrossRef Soriano P, Gendreau M (1996) Diversification strategies in tabu search algorithms for the maximum clique problem. Ann Oper Res 63(2):189–207CrossRef
167.
Zurück zum Zitat Soriano P, Gendreau M (1996) Tabu search algorithms for the maximum clique problem. In: Johnson DS, Trick MA (eds) Cliques, coloring, and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence, pp 221–242CrossRef Soriano P, Gendreau M (1996) Tabu search algorithms for the maximum clique problem. In: Johnson DS, Trick MA (eds) Cliques, coloring, and satisfiability: second DIMACS implementation challenge. American Mathematical Society, Providence, pp 221–242CrossRef
168.
Zurück zum Zitat Takefuji Y, Lee KC (1991) Artificial neural networks for four-coloring map problems and k-colorability problems. IEEE Trans Circuits Syst 38(3):326–333CrossRef Takefuji Y, Lee KC (1991) Artificial neural networks for four-coloring map problems and k-colorability problems. IEEE Trans Circuits Syst 38(3):326–333CrossRef
169.
Zurück zum Zitat Talaván PM, Yáñez J (2008) The graph coloring problem: a neuronal network approach. Eur J Oper Res 191(1):100–111MathSciNetCrossRef Talaván PM, Yáñez J (2008) The graph coloring problem: a neuronal network approach. Eur J Oper Res 191(1):100–111MathSciNetCrossRef
170.
Zurück zum Zitat Titiloye O, Crispin A (2011) Graph coloring with a distributed hybrid quantum annealing algorithm. In: Agent and multi-agent systems: technologies and applications. Springer, Berlin/Heidelberg, pp 553–562CrossRef Titiloye O, Crispin A (2011) Graph coloring with a distributed hybrid quantum annealing algorithm. In: Agent and multi-agent systems: technologies and applications. Springer, Berlin/Heidelberg, pp 553–562CrossRef
171.
172.
Zurück zum Zitat Verma A, Buchanan A, Butenko S (2015) Solving the maximum clique and vertex coloring problems on very large sparse networks. INFORMS J Comput 27(1):164–177CrossRef Verma A, Buchanan A, Butenko S (2015) Solving the maximum clique and vertex coloring problems on very large sparse networks. INFORMS J Comput 27(1):164–177CrossRef
173.
Zurück zum Zitat Welsh DJ, Powell MB (1967) An upper bound for the chromatic number of a graph and its application to timetabling problems. Comput J 10(1):85–86CrossRef Welsh DJ, Powell MB (1967) An upper bound for the chromatic number of a graph and its application to timetabling problems. Comput J 10(1):85–86CrossRef
175.
Zurück zum Zitat de Werra D (1990) Heuristics for graph coloring. In: Computational graph theory. Springer, Berlin, pp 191–208CrossRef de Werra D (1990) Heuristics for graph coloring. In: Computational graph theory. Springer, Berlin, pp 191–208CrossRef
176.
177.
Zurück zum Zitat Woo TK, Su SY, Newman-Wolfe R (1991) Resource allocation in a dynamically partitionable bus network using a graph coloring algorithm. IEEE Trans Commun 39(12):1794–1801CrossRef Woo TK, Su SY, Newman-Wolfe R (1991) Resource allocation in a dynamically partitionable bus network using a graph coloring algorithm. IEEE Trans Commun 39(12):1794–1801CrossRef
178.
Zurück zum Zitat Wood D (1969) A technique for colouring a graph applicable to large scale timetabling problems. Comput J 12(4):317–319MathSciNetCrossRef Wood D (1969) A technique for colouring a graph applicable to large scale timetabling problems. Comput J 12(4):317–319MathSciNetCrossRef
179.
Zurück zum Zitat Wu Q, Hao JK (2012) Coloring large graphs based on independent set extraction. Comput Oper Res 39(2):283–290MathSciNetCrossRef Wu Q, Hao JK (2012) Coloring large graphs based on independent set extraction. Comput Oper Res 39(2):283–290MathSciNetCrossRef
180.
Zurück zum Zitat Wu Q, Hao JK (2013) An adaptive multistart tabu search approach to solve the maximum clique problem. J Comb Optim 26(1):86–108MathSciNetCrossRef Wu Q, Hao JK (2013) An adaptive multistart tabu search approach to solve the maximum clique problem. J Comb Optim 26(1):86–108MathSciNetCrossRef
181.
Zurück zum Zitat Wu Q, Hao JK (2013) An extraction and expansion approach for graph coloring. Asia Pac J Oper Res 30(05):1350018MathSciNetCrossRef Wu Q, Hao JK (2013) An extraction and expansion approach for graph coloring. Asia Pac J Oper Res 30(05):1350018MathSciNetCrossRef
182.
183.
Zurück zum Zitat Wu Q, Hao JK, Glover F (2012) Multi-neighborhood tabu search for the maximum weight clique problem. Ann Oper Res 196:611–634MathSciNetCrossRef Wu Q, Hao JK, Glover F (2012) Multi-neighborhood tabu search for the maximum weight clique problem. Ann Oper Res 196:611–634MathSciNetCrossRef
184.
Zurück zum Zitat Youssef SM, Elliman DG (2004) Reactive prohibition-based ant colony optimization (rpaco): a new parallel architecture for constrained clique sub-graphs. In: 16th IEEE international conference on tools with artificial intelligence, pp 63–70 Youssef SM, Elliman DG (2004) Reactive prohibition-based ant colony optimization (rpaco): a new parallel architecture for constrained clique sub-graphs. In: 16th IEEE international conference on tools with artificial intelligence, pp 63–70
185.
Zurück zum Zitat Zhang BT, Shin SY (1999) Code optimization for dna computing of maximal cliques. In: Advances in soft computing. Springer, Heidelberg, pp 588–599CrossRef Zhang BT, Shin SY (1999) Code optimization for dna computing of maximal cliques. In: Advances in soft computing. Springer, Heidelberg, pp 588–599CrossRef
186.
Zurück zum Zitat Zhang Q, Sun J, Tsang E (2005) An evolutionary algorithm with guided mutation for the maximum clique problem. IEEE Trans Evol Comput 9(2):192–200CrossRef Zhang Q, Sun J, Tsang E (2005) An evolutionary algorithm with guided mutation for the maximum clique problem. IEEE Trans Evol Comput 9(2):192–200CrossRef
187.
Zurück zum Zitat Zuckerman D (2007) Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput 3:103–128MathSciNetCrossRef Zuckerman D (2007) Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput 3:103–128MathSciNetCrossRef
188.
Zurück zum Zitat Zufferey N (2012) Optimization by ant algorithms: possible roles for an individual ant. Optim Lett 6(5):963–973MathSciNetCrossRef Zufferey N (2012) Optimization by ant algorithms: possible roles for an individual ant. Optim Lett 6(5):963–973MathSciNetCrossRef
Metadaten
Titel
The Maximum Clique and Vertex Coloring
verfasst von
Oleksandra Yezerska
Sergiy Butenko
Copyright-Jahr
2018
DOI
https://doi.org/10.1007/978-3-319-07124-4_47