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Chordal Editing is Fixed-Parameter Tractable

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Abstract

Graph modification problems typically ask for a small set of operations that transforms a given graph to have a certain property. The most commonly considered operations include vertex deletion, edge deletion, and edge addition; for the same property, one can define significantly different versions by allowing different operations. We study a very general graph modification problem that allows all three types of operations: given a graph and integers , and , the chordal editing problem asks whether can be transformed into a chordal graph by at most vertex deletions, edge deletions, and edge additions. Clearly, this problem generalizes both chordal deletion and chordal completion (also known as minimum fill-in). Our main result is an algorithm for chordal editing in time , where and is the number of vertices of . Therefore, the problem is fixed-parameter tractable parameterized by the total number of allowed operations. Our algorithm is both more efficient and conceptually simpler than the previously known algorithm for the special case chordal deletion.

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Notes

  1. Refer to Sect. 6 for more intuition behind this observation.

  2. This can be surely extended to some local clique tree structure, and we use clique tree here for simplicity.

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Acknowledgments

We are grateful to the anonymous referees for their careful reading and helpful suggestions, and in particular for pointing out flaws in a preliminary version and suggesting a better proof for Lemma 4.2.

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Authors and Affiliations

  1. Department of Computing, Hong Kong Polytechnic University, Hong Kong, China

    Yixin Cao

  2. Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI), Budapest, Hungary

    Yixin Cao & Dániel Marx

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  1. Yixin Cao
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  2. Dániel Marx
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Correspondence to Dániel Marx.

Additional information

This article was supported by the European Research Council (ERC) Grant 280152 and the Hungarian Scientific Research Fund (OTKA) Grant NK105645. A preliminary version of this paper appeared in the proceedings of STACS 2014.

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Cao, Y., Marx, D. Chordal Editing is Fixed-Parameter Tractable. Algorithmica 75, 118–137 (2016). https://doi.org/10.1007/s00453-015-0014-x

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