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A polytime preprocess algorithm for the maximum independent set problem

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Abstract

The maximum independent set (MIS) seeks to find a subset of vertices with the maximum size such that no pair of its vertices are adjacent. This paper develops a recursive fixing procedure that generalizes the existing polytime algorithm to solve the maximum independent set problem on chordal graphs, which admit simplicial orderings. We prove that the generalized fixing procedure is safe; i.e., it does not remove all optimal solutions of the MIS problem from the solution space. Our computational results show that the proposed recursive fixing algorithm, along with the basic mixed integer programming (MIP) of the MIS, outperforms the pure MIP formulation of the problem. Our codes, data, and results are available on GitHub.

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Notes

  1. We note that at most one neighbor of u can be selected in an independent set as \(G[N_G(u)]\) forms a clique in G.

  2. It is not perfect because the clique number of the graph, which is 2, does not equal the chromatic number of the graph, which is 3.

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Acknowledgements

This material is partially based upon work supported by Rice University’s Building Research on Inequality and Diversity to Grow Equity (BRIDGE) seed grant. The authors thank the anonymous reviewer for helpful comments.

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Authors and Affiliations

  1. Computational Applied Mathematics and Operations Research, Rice University, Houston, USA

    Samuel Kroger & Illya V. Hicks

  2. Industrial, Manufacturing & Systems Engineering, Texas Tech University, Lubbock, USA

    Hamidreza Validi

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  1. Samuel Kroger
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  2. Hamidreza Validi
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  3. Illya V. Hicks
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Correspondence to Hamidreza Validi.

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Kroger, S., Validi, H. & Hicks, I.V. A polytime preprocess algorithm for the maximum independent set problem. Optim Lett 18, 651–661 (2024). https://doi.org/10.1007/s11590-023-02076-8

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