Abstract
Given an undirected graph G=(V,E) with vertex set V={1,…,n} and edge set E⊆V×V. Let w:V→Z + be a weighting function that assigns to each vertex i∈V a positive integer. The maximum weight clique problem (MWCP) is to determine a clique of maximum weight. This paper introduces a tabu search heuristic whose key features include a combined neighborhood and a dedicated tabu mechanism using a randomized restart strategy for diversification. The proposed algorithm is evaluated on a total of 136 benchmark instances from different sources (DIMACS, BHOSLIB and set packing). Computational results disclose that our new tabu search algorithm outperforms the leading algorithm for the maximum weight clique problem, and in addition rivals the performance of the best methods for the unweighted version of the problem without being specialized to exploit this problem class.
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Notes
The source code of our MN/TS algorithm is publicly available at: http://www.info.univ-angers.fr/pub/hao/clique.html.
References
Alidaee, B., Kochenberger, G. A., Lewis, K., Lewis, M., & Wang, H. (2008). A new approach for modelling and solving set packing problems. European Journal of Operational Research, 86(2), 504–512.
Babel, L. (1994). A fast algorithm for the maximum weight clique problem. Computing, 52, 31–38.
Ballard, D., & Brown, C. (1982). Computer vision. Englewood Cliffs: Prentice-Hall.
Battiti, R., & Mascia, F. (2010). Reactive and dynamic local search for the Max-Clique problem: engineering effective building blocks. Computers & Operations Research, 37, 534–542.
Battiti, R., & Protasi, M. (2001). Reactive local search for the maximum clique problem. Algorithmica, 29(4), 610–637.
Bomze, I. M., Pelillo, M., & Stix, V. (2000). Approximating the maximum weight clique using replicator dynamics. IEEE Transactions on Neural Networks, 11, 1228–1241.
Busygin, S. (2006). A new trust region technique for the maximum weight clique problem. Discrete Applied Mathematics, 154, 2080–2096.
Cai, S., Su, K., & Chen, Q. (2011). Local search with edge weighting and configuration checking heuristics for minimum vertex cover. Artificial Intelligence, 175(9–10), 1672–1696.
Delorme, X., Gandibleux, X., & Rodriguez, J. (2004). GRASP for set packing problems. European Journal of Operational Research, 153, 564–580.
Di Gaspero, L., & Schaerf, A. (2006). Neighborhood portfolio approach for local search applied to timetabling problems. Journal of Mathematical Modelling and Algorithms, 5(1), 65–89.
Friden, C., Hertz, A., & de Werra, D. (1989). Stabulus: a technique for finding stable sets in large graphs with tabu search. Computing, 42, 35–44.
Gendreau, M., Soriano, P., & Salvail, L. (1993). Solving the maximum clique problem using a tabu search approach. Annals of Operations Research, 41, 385–403.
Glover, F., & Laguna, M. (1997). Tabu search. Boston: Kluwer Academic.
Grosso, A., Locatelli, M., & Croce, F. D. (2004). Combining swaps and node weights in an adaptive greedy approach for the maximum clique problem. Journal of Heuristics, 10, 135–152.
Johnson, D. S., & Trick, M. A. (Eds.) (1996). Cliques, coloring, and satisfiability: second DIMACS Implementation Challenge. DIMACS series in discrete mathematics and theoretical computer science (Vol. 26). Providence: AMS.
Karp, R. M. (1972). Reducibility among combinatorial problems. In R. E. Miller & J. W. Thatcher (Eds.), Complexity of computer computations (pp. 85–103). New York: Plenum Press.
Katayama, K., Hamamoto, A., & Narihisa, H. (2005). An effective local search for the maximum clique problem. Information Processing Letters, 95, 503–511.
Kwon, R. H. (2005). Data dependent worst case bounds for weighted set packing. European Journal of Operational Research, 167(1), 68–76.
Lü, Z., Hao, J. K., & Glover, F. (2011). Neighborhood analysis: a case study on curriculum-based course timetabling. Journal of Heuristics, 17(2), 97–118.
Mannino, C., & Stefanutti, E. (1999). An augmentation algorithm for the maximum weighted stable set problem. Computational Optimization and Applications, 14, 367–381.
Östergård, P. R. J. (2001). A new algorithm for the maximum weight clique problem. Nordic Journal of Computing, 8, 424–436.
Pullan, W. (2008). Approximating the maximum vertex/edge weighted clique using local search. Journal of Heuristics, 14(2), 117–134.
Pullan, W., & Hoos, H. H. (2006). Dynamic local search for the maximum clique problem. The Journal of Artificial Intelligence Research, 25, 159–185.
Wu, Q., & Hao, J. K. (2012). An adaptive multistart tabu search approach to solve the maximum clique problem. Journal of Combinatorial Optimization. doi:10.1007/s10878-011-9437-8.
Acknowledgements
We are grateful to the referees for their suggestions. The work is partially supported by the “Pays de la Loire” Region (France) within the RaDaPop (2009-2013) and LigeRO (2010-2013) projects.
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Wu, Q., Hao, JK. & Glover, F. Multi-neighborhood tabu search for the maximum weight clique problem. Ann Oper Res 196, 611–634 (2012). https://doi.org/10.1007/s10479-012-1124-3
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DOI: https://doi.org/10.1007/s10479-012-1124-3