Exact exponential-time algorithms for finding bicliques

Daniel Binkele-Raible (1) , Henning Fernau (1) , Serge Gaspers (2) , Mathieu Liedloff (3)

1 Trier University
2 CMM - Centre de modélisation mathématique / Centro de Modelamiento Matemático [Santiago]
3 LIFO - Laboratoire d'Informatique Fondamentale d'Orléans

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 $X, Y \subseteq V$, $X\cap Y=\emptyset$, is a non-induced biclique if $\{x,y\}\in E$ for any $x \in X$ and $y \in Y$. 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 $O(1.30052^n)$.

Domaines: Algorithme et structure de données [cs.DS]

Daniel Binkele-Raible, Henning Fernau, Serge Gaspers, Mathieu Liedloff. Exact exponential-time algorithms for finding bicliques. Information Processing Letters, 2010, 111, pp.64-67. ⟨10.1016/j.ipl.2010.10.020⟩. ⟨hal-00535626⟩