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On the Approximate Compressibility of Connected Vertex Cover

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Abstract

The Connected Vertex Cover problem, where the goal is to compute a minimum set of vertices in a given graph which forms a vertex cover and induces a connected subgraph, is a fundamental combinatorial problem and has received extensive attention in various subdomains of algorithmics. In the area of kernelization, it is known that this problem is unlikely to have a polynomial kernelization algorithm. However, it has been shown in a recent work of Lokshtanov et al. (Proceedings of the 49th annual ACM SIGACT symposium on theory of computing, STOC 2017, Montreal, QC, Canada, 2017) that if one considered an appropriate notion of approximate kernelization, then this problem parameterized by the solution size does admit an approximate polynomial kernelization. In fact, Lokshtanov et al. were able to obtain a polynomial size approximate kernelization scheme (PSAKS) for Connected Vertex Cover parameterized by the solution size. A PSAKS is essentially a preprocessing algorithm whose error can be made arbitrarily close to 0. In this paper we revisit this problem, and consider parameters that are strictly smaller than the size of the solution and obtain the first polynomial size approximate kernelization schemes for the Connected Vertex Cover problem when parameterized by the deletion distance of the input graph to the class of cographs, the class of bounded treewidth graphs, and the class of all chordal graphs.

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Authors and Affiliations

  1. Royal Holloway, University of London, Egham, UK

    Diptapriyo Majumdar

  2. University of Warwick, Coventry, UK

    M. S. Ramanujan

  3. The Institute of Mathematical Sciences, HBNI, Chennai, India

    Saket Saurabh

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  1. Diptapriyo Majumdar
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  2. M. S. Ramanujan
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  3. Saket Saurabh
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Correspondence to Diptapriyo Majumdar.

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A part of this work was done when the first author was affiliated to The Institute of Mathematical Sciences, HBNI, Chennai, India.

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Majumdar, D., Ramanujan, M.S. & Saurabh, S. On the Approximate Compressibility of Connected Vertex Cover. Algorithmica 82, 2902–2926 (2020). https://doi.org/10.1007/s00453-020-00708-4

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