An Exact Algorithm for the Minimum Dominating Clique Problem

Dieter Kratsch (1) , Mathieu Liedloff (1)

1 LITA - Laboratoire d'Informatique Théorique et Appliquée

A subset of vertices $D \subseteq V$ of a graph G = (V,E) is a dominating clique if D is a dominating set and a clique of G. The existence problem "Given a graph G, is there a dominating clique in G?" is NP-complete, and thus both the Minimum and the Maximum Dominating Clique problem are NP-hard. We present an $O(1.3390^n)$ time algorithm that for an input graph on n vertices either computes a minimum dominating clique or reports that the graph has no dominating clique. The algorithm uses the Branch & Reduce paradigm and its time analysis is based on the Measure & Conquer approach. We also establish a lower bound of $\Omega(1.2599^n)$ for the worst case running time of our algorithm.

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

Dieter Kratsch, Mathieu Liedloff. An Exact Algorithm for the Minimum Dominating Clique Problem. IWPEC'2006 : 2nd International Workshop on Parameterized and Exact Computation, Sep 2006, Zürich, Switzerland. pp.130-141, ⟨10.1007/11847250_12⟩. ⟨hal-00460727⟩