Parameterized Algorithms and Hardness Results for Some Graph Motif Problems

  • Authors:

  • Nadja Betzler;Michael R. Fellows;Christian Komusiewicz;Rolf Niedermeier

  • Affiliations:

  • Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany D-07743;PC Research Unit, Office of DVC (Research), University of Newcastle, Callaghan, Australia NSW 2308;Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany D-07743;Institut für Informatik, Friedrich-Schiller-Universität Jena, Jena, Germany D-07743

  • Venue:
  • CPM '08 Proceedings of the 19th annual symposium on Combinatorial Pattern Matching
  • Year:
  • 2008

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Abstract

We study the NP-complete Graph Motifproblem: given a vertex-colored graph G= (V,E) and a multiset Mof colors, does there exist an S茂戮驴 Vsuch that G[S] is connected and carries exactly (also with respect to multiplicity) the colors in M? We present an improved randomized algorithm for Graph Motifwith running time O(4.32|M|·|M|2·|E|). We extend our algorithm to list-colored graph vertices and the case where the motif G[S] needs not be connected. By way of contrast, we show that extending the request for motif connectedness to the somewhat "more robust" motif demands of biconnectedness or bridge-connectedness leads to W[1]-complete problems. Actually, we show that the presumably simpler problems of finding (uncolored) biconnected or bridge-connected subgraphs are W[1]-complete with respect to the subgraph size. Answering an open question from the literature, we further show that the parameter "number of connected motif components" leads to W[1]-hardness even when restricted to graphs that are paths.