Tabu search with feasible and infeasible searches for equitable coloring

Abstract The equitable coloring problem is a variant of the classical graph coloring problem that arises from a number of real-life applications where the cardinality of color classes must be balanced. In this paper, we present a highly effective hybrid tabu search method for the problem. Based on three complementary neighborhoods, the algorithm alternates between a feasible local search phase where the search focuses on the most relevant feasible solutions and an infeasible local search phase where a controlled exploration of infeasible solutions is allowed by relaxing the equity constraint. A novel cyclic exchange neighborhood is also proposed in order to enhance the search ability of the hybrid tabu search algorithm. Experiments on a set of 73 benchmark instances in the literature indicate that the proposed algorithm is able to find improved best solutions for 15 instances (new upper bounds) and matches the best-known solutions for 57 instances. Additional analyses show the interest of the cyclic exchange neighborhood and the hybrid scheme combining both feasible and infeasible local searches.

[1]  Thomas A. Runkler,et al.  Particle swarm optimization for generating interpretable fuzzy reinforcement learning policies , 2016, Eng. Appl. Artif. Intell..

[2]  Daniel Brélaz,et al.  New methods to color the vertices of a graph , 1979, CACM.

[3]  Fred W. Glover,et al.  A user's guide to tabu search , 1993, Ann. Oper. Res..

[4]  Fedor V. Fomin,et al.  Equitable Colorings of Bounded Treewidth Graphs , 2004, MFCS.

[5]  Jin-Kao Hao,et al.  Hybrid Evolutionary Algorithms for Graph Coloring , 1999, J. Comb. Optim..

[6]  Manoj Kumar Tiwari,et al.  Composite particle algorithm for sustainable integrated dynamic ship routing and scheduling optimization , 2016, Comput. Ind. Eng..

[7]  Pablo San Segundo Carrillo.  A new DSATUR-based algorithm for exact vertex coloring , 2011 .

[8]  Graciela L. Nasini,et al.  A Tabu Search Heuristic for the Equitable Coloring Problem , 2014, ISCO.

[9]  Ko-Wei Lih,et al.  Equitable Coloring of Graphs , 1998 .

[10]  Jin-Kao Hao,et al.  An evolutionary approach with diversity guarantee and well-informed grouping recombination for graph coloring , 2010, Comput. Oper. Res..

[11]  Fred W. Glover,et al.  A hybrid metaheuristic approach for the capacitated arc routing problem , 2016, Eur. J. Oper. Res..

[12]  David H. Wolpert,et al.  No free lunch theorems for optimization , 1997, IEEE Trans. Evol. Comput..

[13]  Ko-Wei Lih,et al.  On equitable coloring of bipartite graphs , 1996, Discret. Math..

[14]  Sandy Irani,et al.  Scheduling with conflicts, and applications to traffic signal control , 1996, SODA '96.

[15]  Alexandr V. Kostochka,et al.  Equitable colorings of outerplanar graphs , 2002, Discret. Math..

[16]  J. R. Brown.  Chromatic Scheduling and the Chromatic Number Problem , 1972 .

[17]  Angappa Gunasekaran,et al.  Sustainable maritime inventory routing problem with time window constraints , 2017, Eng. Appl. Artif. Intell..

[18]  Celso C. Ribeiro,et al.  A branch-and-cut algorithm for the equitable coloring problem using a formulation by representatives , 2014, Discret. Appl. Math..

[19]  Alain Hertz,et al.  A survey of local search methods for graph coloring , 2004, Comput. Oper. Res..

[20]  Maysam F. Abbod,et al.  A new MIMO ANFIS-PSO based NARMA-L2 controller for nonlinear dynamic systems , 2017, Eng. Appl. Artif. Intell..

[21]  Graciela L. Nasini,et al.  A polyhedral approach for the equitable coloring problem , 2014, Discret. Appl. Math..

[22]  Jin-Kao Hao,et al.  A memetic algorithm for graph coloring , 2010, Eur. J. Oper. Res..

[23]  Jin-Kao Hao,et al.  A memetic algorithm for the Minimum Sum Coloring Problem , 2013, Comput. Oper. Res..

[24]  P. Lax.  Proof of a conjecture of P. Erdös on the derivative of a polynomial , 1944 .

[25]  A. Tucker,et al.  Perfect Graphs and an Application to Optimizing Municipal Services , 1973 .

[26]  Alexandr V. Kostochka,et al.  On equitable Delta-coloring of graphs with low average degree , 2005, Theor. Comput. Sci..

[27]  Tuan Hoang Anh,et al.  A Novel Particle Swarm Optimization-Based Algorithm for the Graph Coloring Problem , 2009, 2009 International Conference on Information Engineering and Computer Science.

[28]  Edward C. Sewell,et al.  An improved algorithm for exact graph coloring , 1993, Cliques, Coloring, and Satisfiability.

[29]  Saïd Hanafi,et al.  A hybrid quantum particle swarm optimization for the Multidimensional Knapsack Problem , 2016, Eng. Appl. Artif. Intell..

[30]  Jin-Kao Hao,et al.  General swap-based multiple neighborhood tabu search for the maximum independent set problem , 2015, Eng. Appl. Artif. Intell..

[31]  Graciela L. Nasini,et al.  A DSATUR-based algorithm for the Equitable Coloring Problem , 2015, Comput. Oper. Res..

[32]  Marek Kubale,et al.  The Complexity of Equitable Vertex Coloring of Graphs , 2005 .

[33]  Isabel Méndez-Díaz,et al.  A Branch-and-Cut algorithm for graph coloring , 2006, Discret. Appl. Math..

[34]  Carlos A. Coello Coello,et al.  Useful Infeasible Solutions in Engineering Optimization with Evolutionary Algorithms , 2005, MICAI.

[35]  Fred W. Glover,et al.  The case for strategic oscillation , 2011, Ann. Oper. Res..

[36]  Roberto Aringhieri,et al.  Simple but effective heuristics for the 2-constraint bin packing problem , 2018, J. Heuristics.

[37]  Hsing-Chih Tsai,et al.  Unified particle swarm delivers high efficiency to particle swarm optimization , 2017, Appl. Soft Comput..

[38]  Ko-Wei Lih,et al.  Equitable Coloring and the Maximum Degree , 1994, Eur. J. Comb..

[39]  Fred W. Glover,et al.  An evolutionary path relinking approach for the quadratic multiple knapsack problem , 2016, Knowl. Based Syst..

[40]  Fred W. Glover,et al.  Backtracking based iterated tabu search for equitable coloring , 2015, Eng. Appl. Artif. Intell..

[41]  Jin-Kao Hao,et al.  Tabu Search for Graph Coloring, T-Colorings and Set T-Colorings , 1999 .

[42]  T. C. Edwin Cheng,et al.  Hybridization of tabu search with feasible and infeasible local searches for the quadratic multiple knapsack problem , 2016, Comput. Oper. Res..

[43]  A. V. Kostochkaa,et al.  On equitable -coloring of graphs with low average degree , 2005 .

[44]  R. Moeini,et al.  Constrained gravitational search algorithm for large scale reservoir operation optimization problem , 2017, Eng. Appl. Artif. Intell..

[45]  Ko-Wei Lih,et al.  Equitable Coloring of Trees , 1994, J. Comb. Theory, Ser. B.

[46]  Victor A. Campos,et al.  On the asymmetric representatives formulation for the vertex coloring problem , 2005, Discret. Appl. Math..

[47]  Alain Hertz,et al.  Using tabu search techniques for graph coloring , 1987, Computing.