Abstract
The advent of desktop multi-core computers has dramatically improved the usability of parallel algorithms which, in the past, have required specialised hardware. This paper introduces cooperating local search (CLS), a parallelised hyper-heuristic for the maximum clique problem. CLS utilises cooperating low level heuristics which alternate between sequences of iterative improvement, during which suitable vertices are added to the current clique, and plateau search, where vertices of the current clique are swapped with vertices not in the current clique. These low level heuristics differ primarily in their vertex selection techniques and their approach to dealing with plateaus. To improve the performance of CLS, guidance information is passed between low level heuristics directing them to particular areas of the search domain. In addition, CLS dynamically reconfigures the allocation of low level heuristics to cores, based on information obtained during a trial, to ensure that the mix of low level heuristics is appropriate for the instance being optimised. CLS has no problem instance dependent parameters, improves the state-of-the-art performance for the maximum clique problem over all the BHOSLIB benchmark instances and attains unprecedented consistency over the state-of-the-art on the DIMACS benchmark instances.
Similar content being viewed by others
References
Balus, E., Yu, C.: Finding a maximum clique in an arbitrary graph. SIAM J. Comput. 15(4), 1054–1068 (1986)
Battiti, R., Protasi, M.: Reactive local search for the maximum clique problem. Algorithmica 29, 610–637 (2001)
Bomze, I., Budinich, M., Pardalos, P., Pelillo, M.: The maximum clique problem. In: Du, D.Z., Pardalos, P. (eds.) Handbook of Combinatorial Optimization, vol. A, pp. 1–74. Kluwer Academic, Norwell (1999)
Boppana, R., Halldórsson, M.: Approximating maximum independent sets by excluding subgraphs. BIT 32, 180–196 (1992)
Brockington, M., Culberson, J.: Camouflaging independent sets in quasi-random graphs. In: Johnson, D.S., Trick, M. (eds.) Cliques, Coloring and Satisfiability: Second DIMACS Implementation Challenge. DIMACS Series, vol. 26. American Mathematical Society, Providence (1996)
Burke, E., Hart, E., Kendall, G., Newall, J., Ross, P., Schulenburg, S.: Hyper-heuristics: An emerging direction in modern search technology. In: Glover, F. (ed.) Handbook of Meta-heuristics, pp. 457–474. Kluwer Academic, Norwell (2003)
Burns, G., Daoud, R., Vaigl, J.: LAM: An open cluster environment for MPI. In: Proceedings of Supercomputing Symposium, pp. 379–386. ACM, New York (1994)
Busygin, S.: A new trust region technique for the maximum weight clique problem. Discrete Appl. Math. 154(15), 2080–2096 (2006)
Chakhlevitch, K., Cowling, P.: Hyper-heuristics: Recent developments. In: Cotta, C., Sevaux, M., Sörensen, K. (eds.): Adaptive and Multilevel Metaheuristics. Studies in Computational Intelligence, vol. 136, pp. 3–29. Springer, Berlin (2008)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of \(\mathcal{NP}\)-Completeness. Freeman, San Francisco (1979)
Grosso, A., Locatelli, M., Croce, F.D.: Combining swaps and node weights in an adaptive greedy approach for the maximum clique problem. J. Heuristics 10, 135–152 (2004)
Grosso, A., Locatelli, M., Pullan, W.: Randomness, plateau search, penalties, restart rules: simple ingredients leading to very efficient heuristics for the maximum clique problem. J. Heuristics 14, 587–612 (2008)
Hansen, P., Mladenović, N., Urosević, D.: Variable neighborhood search for the maximum clique. Discrete Appl. Math. 145, 117–125 (2004)
Håstad, J.: Clique is hard to approximate within n 1−ε. Acta Math. 182, 105–142 (1999)
Ji, Y., Xu, X., Stormo, G.D.: A graph theoretical approach for predicting common RNA secondary structure motifs including pseudoknots in unaligned sequences. Bioinformatics 20(10), 1591–1602 (2004)
Johnson, D., Trick, M. (eds.): Cliques, Coloring and Satisfiability: Second DIMACS Implementation Challenge. DIMACS Series, vol. 26. American Mathematical Society, Providence (1996)
Katayama, K., Hamamoto, A., Narihisa, H.: Solving the maximum clique problem by k-opt local search. In: Proceedings of the 2004 ACM Symposium on Applied Computing, pp. 1021–1025. ACM, New York (2004)
Marchiori, E.: Genetic, iterated and multistart local search for the maximum clique problem. In: Applications of Evolutionary Computing. Lecture Notes in Computer Science, vol. 2279, pp. 112–121. Springer, Berlin (2002)
Pevzner, P.A., Sze, S.H.: Combinatorial approaches to finding subtle signals in DNA sequences. In: Proceedings of the Eighth International Conference on Intelligent Systems for Molecular Biology, pp. 269–278. AAAI Press, Menlo Park (2000)
Pullan, W.: Phased local search for the maximum clique problem. J. Combin. Optim. 12, 303–323 (2006)
Pullan, W.: Approximating the maximum vertex/edge weighted clique using local search. J. Heuristics 14, 117–134 (2008)
Pullan, W., Hoos, H.: Dynamic local search for the maximum clique problem. J. Artif. Intell. Res. 25, 159–185 (2006)
Resende, M., Feo, T., Smith, S.: Algorithm 786: FORTRAN subroutine for approximate solution of the maximum independent set problem using GRASP. ACM Trans. Math. Softw. 24, 386–394 (1998)
Squyres, J.M., Lumsdaine, A.: A component architecture for LAM/MPI. In: Proceedings, 10th European PVM/MPI Users’ Group Meeting, Venice, Italy, September/October 2003. Lecture Notes in Computer Science, vol. 2840, pp. 379–387. Springer, Berlin (2003)
Wolpert, D., Macready, G.: No free lunch theorems for optimization. IEEE Trans. Evol. Comput. 1, 67–82 (1997)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pullan, W., Mascia, F. & Brunato, M. Cooperating local search for the maximum clique problem. J Heuristics 17, 181–199 (2011). https://doi.org/10.1007/s10732-010-9131-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10732-010-9131-5