Fast local search for the maximum independent set problem
- 572 Downloads
Given a graph G=(V,E), the independent set problem is that of finding a maximum-cardinality subset S of V such that no two vertices in S are adjacent. We introduce two fast local search routines for this problem. The first can determine in linear time whether a maximal solution can be improved by replacing a single vertex with two others. The second routine can determine in O(mΔ) time (where Δ is the highest degree in the graph) whether there are two solution vertices than can be replaced by a set of three. We also present a more elaborate heuristic that successfully applies local search to find near-optimum solutions to a wide variety of instances. We test our algorithms on instances from the literature as well as on new ones proposed in this paper.
KeywordsMaximum independent set Local search Iterated local search Algorithm engineering
We thank Diego Nehab and Pedro Sander for sharing their paper and providing us with the MESH instances. We are also grateful to Manuel Holtgrewe for pointing out a few minor issues with the version of our code we used in the preliminary version of this paper (Andrade et al. 2008). In particular, some vertex scans of ILS were not counted properly, which affected its stopping criterion.
- Andrade, D.V., Resende, M.G.C., Werneck, R.F.: Fast local search for the maximum independent set problem. In: McGeoch, C.C. (ed.) Proc. 7th International Workshop on Experimental Algorithms (WEA), vol. 5038, pp. 220–234. Springer, Berlin (2008) Google Scholar
- Bomze, I.M., Budinich, M., Pardalos, P.M., Pelillo, M.: The maximum clique problem. In: Du, D.Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization (Sup. Vol. A), pp. 1–74. Kluwer Academic, Norwell (1999) Google Scholar
- Butenko, S., Pardalos, P.M., Sergienko, I., Shylo, V., Stetsyuk, P.: Estimating the size of correcting codes using extremal graph problems. In: Pearce, C. (ed.) Optimization: Structure and Applications. Springer, Berlin (2008) Google Scholar
- Demetrescu, C., Goldberg, A.V., Johnson, D.S.: 9th DIMACS Implementation Challenge: Shortest Paths (2006). http://www.dis.uniroma1.it/~challenge9. Last visited on March 15, 2008
- Lourenço, H.R., Martin, O., Stützle, T.: Iterated local search. In: Glover, F., Kochenberger, G. (eds.) Handbook of Metaheuristics, pp. 321–353. Kluwer Academic, Norwell (2003) Google Scholar
- Richter, S., Helmert, M., Gretton, C.: A stochastic local search approach to vertex cover. In: Proceedings of the 30th German Conference on Artificial Intelligence (KI), pp. 412–426 (2007) Google Scholar
- Sloane, N.J.A.: Challenge problems: Independent sets in graphs (2000). http://www.research.att.com/~njas/doc/graphs.html. Last visited on March 15, 2008
- Xu, K.: BHOSLIB: Benchmarks with hidden optimum solutions for graph problems (2004). http://www.nlsde.buaa.edu.cn/~kexu/benchmarks/graph-benchmarks.htm. Last visited on March 15, 2008