Efficient algorithms for finding critical subgraphs

This paper presents algorithms to find vertex-critical and edge-critical subgraphs in a given graph G, and demonstrates how these critical subgraphs can be used to determine the chromatic number of G. Computational experiments are reported on random and DIMACS benchmark graphs to compare the proposed algorithms, as well as to find lower bounds on the chromatic number of these graphs. We improve the best known lower bound for some of these graphs, and we are even able to determine the chromatic number of some graphs for which only bounds were known.

[1]  D. de Werra,et al.  Heuristics for graph coloring , 1989 .

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

[3]  David S. Johnson,et al.  Dimacs series in discrete mathematics and theoretical computer science , 1996 .

[4]  Fred Glover,et al.  Tabu Search - Part II , 1989, INFORMS J. Comput..

[5]  Alan K. Mackworth.  Constraint Satisfaction , 1985 .

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

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

[8]  John W. Chinneck,et al.  Feasibility and Viability , 1997 .

[9]  Joyce van Loon.  Irreducibly inconsistent systems of linear inequalities , 1981 .

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

[11]  Michael A. Trick,et al.  A Column Generation Approach for Graph Coloring , 1996, INFORMS J. Comput..

[12]  Alain Hertz,et al.  Finding the chromatic number by means of critical graphs , 2000, Electron. Notes Discret. Math..

[13]  Edward P. K. Tsang,et al.  Foundations of constraint satisfaction , 1993, Computation in cognitive science.

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

[15]  Alain Hertz,et al.  Solution techniques for the Large Set Covering Problem , 2003, Discret. Appl. Math..

[16]  John W. Chinneck,et al.  Finding a Useful Subset of Constraints for Analysis in an Infeasible Linear Program , 1997, INFORMS J. Comput..

[17]  Fred W. Glover,et al.  Tabu Search - Part I , 1989, INFORMS J. Comput..

[18]  Alejandro Pazos Sierra,et al.  Encyclopedia of Artificial Intelligence , 2008 .

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

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

[21]  A. Mehrotra,et al.  A column generation approach for exact graph coloring , 1994 .

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

[23]  Marek Kubale,et al.  A generalized implicit enumeration algorithm for graph coloring , 1985, CACM.

[24]  Jürgen Peemöller,et al.  A correction to Brelaz's modification of Brown's coloring algorithm , 1983, CACM.

[25]  Alain Hertz,et al.  An adaptive memory algorithm for the k-coloring problem , 2003, Discret. Appl. Math..

[26]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .