Exact exponential-time algorithms for finding bicliques

  • Authors:

  • Daniel Binkele-Raible;Henning Fernau;Serge Gaspers;Mathieu Liedloff

  • Affiliations:

  • Universität Trier, FB 4 - Abteilung Informatik, D-54286 Trier, Germany;Universität Trier, FB 4 - Abteilung Informatik, D-54286 Trier, Germany;Institute of Information Systems, Vienna University of Technology, Vienna, Austria;LIFO, Université d'Orléans, 45067 Orléans Cedex 2, France

  • Venue:
  • Information Processing Letters
  • Year:
  • 2010

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Abstract

Due to a large number of applications, bicliques of graphs have been widely considered in the literature. This paper focuses on non-induced bicliques. Given a graph G=(V,E) on n vertices, a pair (X,Y), with X,Y@?V, X@?Y=@A, is a non-induced biclique if {x,y}@?E for any x@?X and y@?Y. The NP-complete problem of finding a non-induced (k"1,k"2)-biclique asks to decide whether G contains a non-induced biclique (X,Y) such that |X|=k"1 and |Y|=k"2. In this paper, we design a polynomial-space O(1.6914^n)-time algorithm for this problem. It is based on an algorithm for bipartite graphs that runs in time O(1.30052^n). In deriving this algorithm, we also exhibit a relation to the spare allocation problem known from memory chip fabrication. As a byproduct, we show that the constraint bipartite vertex cover problem can be solved in time O(1.30052^n).