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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.

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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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Authors and Affiliations

  1. Google Inc., 76 Ninth Avenue, New York, NY, 10011, USA

    Diogo V. Andrade

  2. AT&T Labs Research, 180 Park Ave, Florham Park, NJ, 07932, USA

    Mauricio G. C. Resende

  3. Microsoft Research Silicon Valley, 1065 La Avenida, Mtn. View, CA, 94043, USA

    Renato F. Werneck

Authors
  1. Diogo V. Andrade
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  2. Mauricio G. C. Resende
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  3. Renato F. Werneck
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Corresponding author

Correspondence to Mauricio G. C. Resende.

Additional information

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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