Using Constraint Programming to Solve the Maximum Clique Problem
This paper aims to show that Constraint Programming can be an efficient technique to solve a well-known combinatorial optimization problem: the search for a maximum clique in a graph. A clique of a graph G=(X,E) is a subset V of X, such that every two nodes in V are joined by an edge of E. The maximum clique problem consists of finding ω(G) the largest cardinality of a clique. We propose two new upper bounds of ω(G) and a new strategy to guide the search for an optimal solution. The interest of our approach is emphasized by the results we obtain for the DIMACS Benchmarks. Seven instances are solved for the first time and two better lower bounds for problems remaining open are found. Moreover, we show that the CP method we propose gives good results and quickly.
Unable to display preview. Download preview PDF.
- 1.Balas, E., Niehaus, W.: Finding large cliques in arbitrary graphs by bipartite matching. In: Johnson, D., Trick, M. (eds.) DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 26, pp. 29–52. American Mathematical Society, Providence (1996)Google Scholar
- 2.Berge, C.: Graphe et Hypergraphes. Dunod, Paris (1970)Google Scholar
- 3.Bomze, I., Budinich, M., Pardalos, P., Pelillo, M.: The maximum clique problem. Handbook of Combinatorial Optimization 4 (1999)Google Scholar
- 5.Busygin, S.: A new trust region technique for the maximum weight clique problem. Submitted to Special Issue of Discrete Applied Mathematics: Combinatorial Optimization (2002)Google Scholar
- 6.Dimacs. Dimacs clique benchmark instances (1993), ftp://dimacs.rutgers.edu/pub/challenge/graph/benchmarks/clique
- 8.Homer, S., Peinado, M.: Experiements with polynomial-time clique approximation algorithms on very large graphs. In: Johnson, D., Trick, M. (eds.) DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 26, pp. 147–168. American Mathematical Society, Providence (1996)Google Scholar
- 9.Östegard, P.: A fast algorithm for the maximum clique problem. Discrete Applied Mathematics (page to appear)Google Scholar
- 10.St-Louis, P., Gendron, B., Ferland, J.: A penalty-evaporation heuristic in a decomposition method for the maximum clique problem. In: Optimization Days, Montreal, Canada (2003)Google Scholar