Parameterizing Edge Modification Problems Above Lower Bounds

  • René van BevernEmail author
  • Vincent Froese
  • Christian Komusiewicz
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)


For a fixed graph F, we study the parameterized complexity of a variant of the \(F\text {-}{\textsc {free\ Editing}}\) 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 show fixed-parameter tractability with respect to \(\ell \) for \(K_3\text {-}\textsc {Free\ Editing}\), Feedback Arc Set in Tournaments, and Cluster Editing when the packing \(\mathcal H\) contains subgraphs with bounded solution size. For \(K_3\text {-}\textsc {Free\ Editing}\), we also prove NP-hardness in case of edge-disjoint packings of \(K_3\)s and \(\ell =0\), while for \(K_q\text {-}\textsc {Free\ Editing}\) and \(q\ge 6\), NP-hardness for \(\ell =0\) even holds for vertex-disjoint packings of \(K_q\)s.


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


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • René van Bevern
    • 1
    Email author
  • Vincent Froese
    • 2
  • Christian Komusiewicz
    • 3
  1. 1.Novosibirsk State UniversityNovosibirskRussian Federation
  2. 2.Technische Universität BerlinBerlinGermany
  3. 3.Friedrich-Schiller-Universität JenaJenaGermany

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