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Theory of Computing Systems

, Volume 62, Issue 3, pp 739–770 | Cite as

Parameterizing Edge Modification Problems Above Lower Bounds

  • René van Bevern
  • Vincent Froese
  • Christian Komusiewicz
Article

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 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 = 0, while for K q -Free Editing and q ≥ 6, NP-hardness for = 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.

Keywords

NP-hard problem Fixed-parameter algorithm Subgraph packing Kernelization Graph-based Clustering Feedback arc set Cluster editing 

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Novosibirsk State UniversityNovosibirskRussian Federation
  2. 2.Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of SciencesNovosibirskRussian Federation
  3. 3.Technische Universität BerlinBerlinGermany
  4. 4.Friedrich-Schiller-Universität JenaJenaGermany

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