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Improved approximation algorithms for hitting 3-vertex paths

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Abstract

We study the problem of deleting a minimum cost set of vertices from a given vertex-weighted graph in such a way that the resulting graph has no induced path on three vertices. This problem is often called cluster vertex deletion in the literature and admits a straightforward 3-approximation algorithm since it is a special case of the vertex cover problem on a 3-uniform hypergraph. Recently, You, Wang, and Cao described an efficient 5/2-approximation algorithm for the unweighted version of the problem. Our main result is a 9/4-approximation algorithm for arbitrary weights, using the local ratio technique. We further conjecture that the problem admits a 2-approximation algorithm and give some support for the conjecture. This is in sharp contrast with the fact that the similar problem of deleting vertices to eliminate all triangles in a graph is known to be UGC-hard to approximate to within a ratio better than 3, as proved by Guruswami and Lee.

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Acknowledgements

We thank the anonymous referees for their careful reading of the paper and helpful remarks.

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Authors and Affiliations

  1. Département de Mathématique, Université Libre de Bruxelles, Brussels, Belgium

    Samuel Fiorini

  2. Département d’Informatique, Université Libre de Bruxelles, Brussels, Belgium

    Gwenaël Joret

  3. Institut für Informatik, Universität zu Köln, Cologne, Germany

    Oliver Schaudt

Authors
  1. Samuel Fiorini
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  2. Gwenaël Joret
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  3. Oliver Schaudt
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Correspondence to Gwenaël Joret.

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S. Fiorini is supported by ERC Consolidator Grant 615640-ForEFront. G. Joret is supported by an ARC grant from the Wallonia-Brussels Federation of Belgium.

A preliminary version of this paper appeared as an extended abstract in the Proceedings of the 18th Conference on Integer Programming and Combinatorial Optimization (IPCO ’16) [5].

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Fiorini, S., Joret, G. & Schaudt, O. Improved approximation algorithms for hitting 3-vertex paths. Math. Program. 182, 355–367 (2020). https://doi.org/10.1007/s10107-019-01395-y

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  • DOI: https://doi.org/10.1007/s10107-019-01395-y

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