An exact algorithm for subset feedback vertex set on chordal graphs

Authors:
Petr A. Golovach;Pinar Heggernes;Dieter Kratsch;Reza Saei
Affiliations:
Department of Informatics, University of Bergen, Norway;Department of Informatics, University of Bergen, Norway;LITA, Université de Lorraine - Metz, France;Department of Informatics, University of Bergen, Norway
Venue:
IPEC'12 Proceedings of the 7th international conference on Parameterized and Exact Computation
Year:
2012

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Abstract

Given a graph G=(V,E) and a set S⊆V, a set U⊆V is a subset feedback vertex set of (G,S) if no cycle in G[V∖U] contains a vertex of S. The Subset Feedback Vertex Set problem takes as input G, S, and an integer k, and the question is whether (G,S) has a subset feedback vertex set of cardinality or weight at most k. Both the weighted and the unweighted versions of this problem are NP-complete on chordal graphs, even on their subclass split graphs. We give an algorithm with running time O(1.6708n) that enumerates all minimal subset feedback vertex sets on chordal graphs with n vertices. As a consequence, Subset Feedback Vertex Set can be solved in time O(1.6708n) on chordal graphs, both in the weighted and in the unweighted case. On arbitrary graphs, the fastest known algorithm for the problems has O(1.8638n) running time.