Simple and Fast: Improving a Branch-And-Bound Algorithm for Maximum Clique
We consider a branch-and-bound algorithm for maximum clique problems. We introduce cost based filtering techniques for the socalled candidate set (i.e. a set of nodes that can possibly extend the clique in the current choice point).
Additionally, we present a taxonomy of upper bounds for maximum clique. Analytical results show that our cost based filtering is in a sense as tight as most of these well-known bounds for the maximum clique problem.
Experiments demonstrate that the combination of cost based filtering and vertex coloring bounds outperforms the old approach as well as approaches that only apply either of these techniques. Furthermore, the new algorithm is competitive with other recent algorithms for maximum clique.
Keywordsmaximum clique branch-and-bound constraint programming cost based filtering
Unable to display preview. Download preview PDF.
- M.R. Garey and D. S. Johnson. Computers and Intractability. W.H. Freeman & Co., 1979.Google Scholar
- E. Balas, S. Ceria, G. Couruéjols and G. Pataki. Polyhedral Methods for the Maximum Clique Problem. in [15, p. 11–28].Google Scholar
- I.M. Bomze, M. Budinich, P.M. Pardalos, M. Pelillo. The Maximum Clique Problem. Handbook of Combinatorial Optimization, volume 4. Kluwer Academic Publishers, 1999.Google Scholar
- T. Fahle. Cost Based Filtering vs. UpperBounds for Maximum Clique CP-AIOR’02 Workshop, Le Croisic/France, 2002.Google Scholar
- T. Fahle and M. Sellmann. Constraint Programming Based Column Generation with Knapsack Subproblems. Annals of Operations Reserach, Vol 114, 2003, to appear.Google Scholar
- F. Focacci, A. Lodi, M. Milano. Cost-Based Domain Filtering. Proc. CP’99 LNCS 1713:189–203, 1999.Google Scholar
- F. Focacci, A. Lodi, M. Milano. Cutting Planes in Constraint Programming: An Hybrid Approach. Proceedings of CP’00, Springer LNCF 1894:187–200, 2000.Google Scholar
- D. S. Johnson and M.A. Trick. Cliques, Colorings and Satisfiability. 2nd DIMACS Implementation Challenge, 1993. American Mathematical Society, 1996.Google Scholar
- U. Junker, S.E. Karisch, N. Kohl, B. Vaaben, T. Fahle, M. Sellmann. A Framework for Constraint programming based column generation. Proc. CP’99 LNCS 1713:261–274, 1999.Google Scholar
- P.R. J. Östergård. A fast algorithm for the maximum clique problem. Discrete Applied Mathematics, to appear.Google Scholar
- G. Ottosson and E. S. Thorsteinsson. Linear Relaxation and Reduced-Cost Based Propagation of Continuous Variable Subscripts. CP-AI-OR’00, Paderborn, 2000, submitted.Google Scholar