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Fast local search for the maximum independent set problem

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Abstract

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.

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Notes

  1. Standard doubly-linked lists will do, but updating them is nontrivial. In particular, when removing a vertex x from the solution, we must be able to remove in constant time the entry representing x in the list of each neighbor v. This can be accomplished by storing a pointer to that entry together with the arc (x,v) in x’s adjacency list.

  2. Due to an implementation issue, a preliminary version of this paper (Andrade et al. 2008) incorrectly stated that the 3-improvement local search could seldom improve 2-maximal solutions. As the tables show, this is not true: 3-improvements are found quite often.

  3. The standard deviation is not always a good measure of regularity. Despite being highly regular, gameguy has a triangulation pattern in which roughly half the vertices have degree 4 and half have degree 8, leading to a standard deviation higher than 2.

References

  • 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 

  • Battiti, R., Protasi, M.: Reactive local search for the maximum clique problem. Algorithmica 29(4), 610–637 (2001)

    Article  MathSciNet  MATH  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.: Finding maximum independent sets in graphs arising from coding theory. In: Proceedings of the 2002 ACM Symposium on Applied Computing, pp. 542–546 (2002)

    Chapter  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

  • Feo, T., Resende, M.G.C., Smith, S.: A greedy randomized adaptive search procedure for maximum independent set. Oper. Res. 42, 860–878 (1994)

    Article  MATH  Google Scholar 

  • Grosso, A., Locatelli, M., Della Croce, F.: Combining swaps and node weights in an adaptive greedy approach for the maximum clique problem. J. Heuristics 10, 135–152 (2004)

    Article  Google Scholar 

  • Grosso, A., Locatelli, M., Pullan, W.: Simple ingredients leading to very efficient heuristics for the maximum clique problem. J. Heuristics 14(6), 587–612 (2008)

    Article  Google Scholar 

  • Hansen, P., Mladenović, N., Urošević, D.: Variable neighborhood search for the maximum clique. Discrete Appl. Math. 145(1), 117–125 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  • Johnson, D.S., Trick, M. (eds.): Cliques, Coloring and Satisfiability. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 26. AMS, Providence (1996)

    MATH  Google Scholar 

  • Karp, R.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum, New York (1972)

    Chapter  Google Scholar 

  • Katayama, K., Hamamoto, A., Narihisa, H.: An effective local search for the maximum clique problem. Inf. Process. Lett. 95, 503–511 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • Pullan, W.J., Hoos, H.H.: Dynamic local search for the maximum clique problem. J. Artif. Intell. Res. 25, 159–185 (2006)

    MATH  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 

  • Sander, P.V., Nehab, D., Chlamtac, E., Hoppe, H.: Efficient traversal of mesh edges using adjacency primitives. ACM Trans. Graph. 27(5), 144:1–144:9 (2008)

    Article  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

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Acknowledgements

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.

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Correspondence to Mauricio G. C. Resende.

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Part of this work was done while D.V. Andrade was at Rutgers University. Part of this work was done while R.F. Werneck was at Princeton University.

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Andrade, D.V., Resende, M.G.C. & Werneck, R.F. Fast local search for the maximum independent set problem. J Heuristics 18, 525–547 (2012). https://doi.org/10.1007/s10732-012-9196-4

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