Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs

Given an undirected graph G = (V,E) and an integer ` ≥ 1, the NPhard 2-Club problem asks for a vertex set S ⊆ V of size at least ` such that the subgraph induced by S has diameter at most two. In this work, we extend previous parameterized complexity studies for 2-Club. On the positive side, we give polynomial-size problem kernels for the parameters feedback edge set size of G and size of a cluster editing set of G and present a direct combinatorial algorithm for the parameter treewidth of G. On the negative side, we first show that unless NP ⊆ coNP/poly, 2-Club does not admit a polynomial-size problem kernel with respect to the size of a vertex cover of G. Next, we show that, under the strong exponential time hypothesis, a previous O(2|V |−` · |V ||E|)-time search tree algorithm [Schäfer et al., Optim. Lett. 2012] cannot be improved and that, unless NP ⊆ coNP/poly, there is no polynomial-size problem kernel for the dual parameter |V | − `. Finally, we show that, in spite of this lower bound, the search tree algorithm for the dual parameter |V | − ` can be tuned into an efficient exact algorithm for 2-Club that outperforms previous implementations. Submitted: July 2013 Reviewed: Revised: Reviewed: Revised: Accepted: Final: Published: Article type: Regular paper Communicated by: A. Editor Research supported by Grant xxxx-xxxx-xxxx E-mail addresses: [email protected] (Sepp Hartung) [email protected] (Christian Komusiewicz) [email protected] (André Nichterlein) JGAA, 0(0) 0–0 (0) 1