A polynomially searchable exponential neighbourhood for graph colouring

In this paper, we develop a new graph colouring strategy. Our heuristic is an example of a so-called polynomially searchable exponential neighbourhood approach. The neighbourhood is that of permutations of the colours of vertices of a subgraph. Our approach provides a solution method for colouring problems with edge weights. Results for initial tests on unweighted K-colouring benchmark problems are presented. Our colour permutation move was found in practice to be too slow to justify its use on these problems. By contrast, our implementation of iterative descent, which incorporates a permutation kickback move, performed extremely well. Moreover, our approach may yet prove valuable for weighted K-colouring. In addition, our approach offers an improved measure of the distance between colourings of a graph.

[1]  Catherine A. Schevon,et al.  Optimization by simulated annealing: An experimental evaluation , 1984 .

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

[3]  Cecilia R. Aragon,et al.  Optimization by Simulated Annealing: An Experimental Evaluation; Part II, Graph Coloring and Number Partitioning , 1991, Oper. Res..

[4]  Feng Luo,et al.  Exploring the k-colorable landscape with Iterated Greedy , 1993, Cliques, Coloring, and Satisfiability.

[5]  David S. Johnson,et al.  Cliques, Coloring, and Satisfiability , 1996 .

[6]  Béla Bollobás,et al.  The chromatic number of random graphs , 1988, Comb..

[7]  Charles Fleurent,et al.  Genetic and hybrid algorithms for graph coloring , 1996, Ann. Oper. Res..

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

[9]  Paolo Toth,et al.  Algorithm 548: Solution of the Assignment Problem [H] , 1980, TOMS.

[10]  Craig A. Morgenstern.  Distributed coloration neighborhood search , 1993, Cliques, Coloring, and Satisfiability.

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

[12]  Cecilia R. Aragon,et al.  Optimization by Simulated Annealing: An Experimental Evaluation; Part I, Graph Partitioning , 1989, Oper. Res..

[13]  Gregory Gutin,et al.  Exponential neighbourhood local search for the traveling salesman problem , 1999, Comput. Oper. Res..

[14]  Celia A. Glass.  Bag rationalisation for a food manufacturer , 2002, J. Oper. Res. Soc..

[15]  Abraham P. Punnen.  The traveling salesman problem: new polynomial approximation algorithms and domination analysis , 2001 .

[16]  Chris N. Potts,et al.  An Iterated Dynasearch Algorithm for the Single-Machine Total Weighted Tardiness Scheduling Problem , 2002, INFORMS J. Comput..

[17]  Gerhard J. Woeginger,et al.  A study of exponential neighborhoods for the Travelling Salesman Problem and for the Quadratic Assignment Problem , 2000, Math. Program..

[18]  Alain Hertz,et al.  Embedding a sequential procedure within an evolutionary algorithm for coloring problems in graphs , 1995, J. Heuristics.