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A tight approximation algorithm for the cluster vertex deletion problem

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Abstract

We give the first 2-approximation algorithm for the cluster vertex deletion problem. This approximation factor is tight, since approximating the problem within any constant factor smaller than 2 is UGC-hard. Our algorithm combines previous approaches, based on the local ratio technique and the management of twins, with a novel construction of a “good” cost function on the vertices at distance at most 2 from any vertex of the input graph. As an additional contribution, we also study cluster vertex deletion from the polyhedral perspective, where we prove almost matching upper and lower bounds on how well linear programming relaxations can approximate the problem.

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Notes

  1. We warn the reader that, in other papers, twins are usually called true twins, whereas two vertices which have the same set of neighbours are called false twins (note that false twins are not adjacent). Since we have no need of false twins in this paper, we have chosen to use twins in place of true twins.

  2. A fan is a graph obtained from a path by adding an apex vertex.

  3. The first inequality follows since \(|H|\leqslant \alpha (H)\cdot \omega (H)\), for every perfect graph H.

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Acknowledgements

We are grateful to Daniel Lokshtanov for suggesting Lemma 4, which allowed us to simplify our algorithm and its proof. We also thank two anonymous referees for their helpful comments, which improved the presentation of the paper.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Padova, Padua, Italy

    Manuel Aprile

  2. Département de Mathématique, Université libre de Bruxelles, Brussels, Belgium

    Matthew Drescher & Samuel Fiorini

  3. School of Mathematics, Monash University, Melbourne, Australia

    Tony Huynh

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  1. Manuel Aprile
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  2. Matthew Drescher
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This paper appeared as an extended abstract in the proceedings of IPCO 2021. See the end of Sect. 1 for a detailed comparison.

This project was supported by ERC Consolidator Grant 615640-ForEFront. Samuel Fiorini and Manuel Aprile are also supported by FNRS grant T008720F-35293308-BD-OCP. Tony Huynh is also supported by the Australian Research Council.

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Aprile, M., Drescher, M., Fiorini, S. et al. A tight approximation algorithm for the cluster vertex deletion problem. Math. Program. 197, 1069–1091 (2023). https://doi.org/10.1007/s10107-021-01744-w

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