Iterated backtrack removal search for finding k-vertex-critical subgraphs

Abstract

Given an undirected graph \(G = (V,E)\) and a positive integer k, a k-vertex-critical subgraph (k-VCS) of G is a subgraph H such that its chromatic number equals k (i.e., \(\chi (H) = k\)), and removing any vertex causes a decrease of \(\chi (H)\). The k-VCS problem (k-VCSP) is to find the smallest k-vertex-critical subgraph \(H^*\) of G. This paper proposes an iterated backtrack-based removal (IBR) heuristic to find k-VCS for a given graph G. IBR extends the popular removal strategy that is intensification-oriented. The proposed extensions include two new diversification-oriented search components—a backtracking mechanism to reconsider some removed vertices and a perturbation strategy to escape local optima traps. Computational results on 80 benchmark graphs show that IBR is very competitive in terms of solution quality and run-time efficiency compared with state-of-the-art algorithms in the literature. Specifically, IBR improves the best-known solutions for 9 graphs and matches the best results for other 70 instances. We investigate the interest of the key components of the proposed algorithm.

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Notes

  1. 1.

    http://www.dimacs.rutgers.edu/.

  2. 2.

    http://www.cs.hbg.psu.edu/txn131/graphcoloring.html/.

  3. 3.

    dfmax: ftp://dimacs.rutgers.edu/pub/dsj/clique/.

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Acknowledgements

We are grateful to Dr. Chumin Li for providing us with the code of Zhou et al. (2014). Support for the first author of this work from the China Scholarship Council (2015–2019) is also acknowledged.

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Correspondence to Jin-Kao Hao.

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Sun, W., Hao, JK. & Caminada, A. Iterated backtrack removal search for finding k-vertex-critical subgraphs. J Heuristics 25, 565–590 (2019). https://doi.org/10.1007/s10732-017-9358-5

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Keywords

  • Vertex-critical subgraph
  • Graph coloring
  • Tabu search
  • Backtracking-based diversification
  • Irreducibly inconsistent systems