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Parameterizing Edge Modification Problems Above Lower Bounds

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Abstract

We study the parameterized complexity of a variant of the F-free Editing problem: Given a graph G and a natural number k, is it possible to modify at most k edges in G so that the resulting graph contains no induced subgraph isomorphic to F? In our variant, the input additionally contains a vertex-disjoint packing \(\mathcal {H}\) of induced subgraphs of G, which provides a lower bound \(h(\mathcal {H})\) on the number of edge modifications required to transform G into an F-free graph. While earlier works used the number k as parameter or structural parameters of the input graph G, we consider instead the parameter \(\ell :=k-h(\mathcal {H})\), that is, the number of edge modifications above the lower bound \(h(\mathcal {H})\). We develop a framework of generic data reduction rules to show fixed-parameter tractability with respect to ℓ for K 3-Free Editing, Feedback Arc Set in Tournaments, and Cluster Editing when the packing \(\mathcal {H}\) contains subgraphs with bounded solution size. For K 3-Free Editing, we also prove NP-hardness in case of edge-disjoint packings of K 3s and ℓ = 0, while for K q -Free Editing and q ≥ 6, NP-hardness for ℓ = 0 even holds for vertex-disjoint packings of K q s. In addition, we provide NP-hardness results for F-free Vertex Deletion, were the aim is to delete a minimum number of vertices to make the input graph F-free.

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Notes

  1. Bounds of this type are exploited, for example, in so-called cutting planes, which are used in speeding up the running time of ILP solvers.

  2. For directed input graphs, we use the term external arc instead of external edge.

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Author information

Authors and Affiliations

  1. Novosibirsk State University, Novosibirsk, Russian Federation

    René van Bevern

  2. Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russian Federation

    René van Bevern

  3. Technische Universität Berlin, Berlin, Germany

    Vincent Froese

  4. Friedrich-Schiller-Universität Jena, Jena, Germany

    Christian Komusiewicz

Authors
  1. René van Bevern
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  2. Vincent Froese
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  3. Christian Komusiewicz
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Correspondence to René van Bevern.

Additional information

An extended abstract of this article appeared in Proceedings of the 11th International Computer Science Symposium in Russia, June 9–13, 2016, St. Petersburg, Russian Federation [6]. René van Bevern is supported by grant 16-31-60007 mol_a_dk of the Russian Foundation for Basic Research. Christian Komusiewicz is supported by grant KO 3669/4-1 of Deutsche Forschungsgemeinschaft.

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van Bevern, R., Froese, V. & Komusiewicz, C. Parameterizing Edge Modification Problems Above Lower Bounds. Theory Comput Syst 62, 739–770 (2018). https://doi.org/10.1007/s00224-016-9746-5

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