# Fast local search for the maximum independent set problem

## Authors

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DOI: 10.1007/s10732-012-9196-4

- Cite this article as:
- Andrade, D.V., Resende, M.G.C. & Werneck, R.F. J Heuristics (2012) 18: 525. doi:10.1007/s10732-012-9196-4

## 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*(*m*Δ) 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.