Covering Graphs with Few Complete Bipartite Subgraphs
Theoretical Computer Science, vol. 410, no. 21-23, pp. 2045-2053, 2009.
A preliminary and shortened version appeared in the proceedings of FSTTCS 2007, the 27th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, Lecture Notes in Computer Science, vol. 4855, pp. 340-351, Springer-Verlag, 2007.
Abstract:We consider computational problems on covering graphs with bicliques (complete bipartite subgraphs). Given a graph and an integer k, the BICLIQUE COVER problem asks whether the edge-set of the graph can be covered with at most k bicliques; the BICLIQUE PARTITION problem is defined similarly with the additional condition that the bicliques are required to be mutually edge-disjoint. The BICLIQUE VERTEX-COVER problem asks whether the vertex-set of the given graph can be covered with at most k bicliques, the BICLIQUE VERTEX-PARTITION problem is defined similarly with the additional condition that the bicliques are required to be mutually vertex-disjoint. All these four problems are known to be NP-complete even if the given graph is bipartite. In this paper we investigate them in the framework of parameterized complexity: do the problems become easier if k is assumed to be small? We show that, considering k as the parameter, the first two problems are fixed-parameter tractable, while the latter two problems are not fixed-parameter tractable unless P=NP.
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