Compression-based fixed-parameter algorithms for feedback vertex set and edge bipartization

Authors:
Jiong Guo;Jens Gramm;Falk Hüffner;Rolf Niedermeier;Sebastian Wernicke
Affiliations:
Institut für Informatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany;Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Sand 13, D-72076 Tübingen, Germany;Institut für Informatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany;Institut für Informatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany;Institut für Informatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, D-07743 Jena, Germany
Venue:
Journal of Computer and System Sciences
Year:
2006

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Abstract

We show that the NP-complete Feedback Vertex Set problem, which asks for the smallest set of vertices to remove from a graph to destroy all cycles, is deterministically solvable in O(c^k@?m) time. Here, m denotes the number of graph edges, k denotes the size of the feedback vertex set searched for, and c is a constant. We extend this to an algorithm enumerating all solutions in O(d^k@?m) time for a (larger) constant d. As a further result, we present a fixed-parameter algorithm with runtime O(2^k@?m^2) for the NP-complete Edge Bipartization problem, which asks for at most k edges to remove from a graph to make it bipartite.