Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs

Given an undirected graph G=(V,E) and an integer l≥1, the NP-hard 2-Club problem asks for a vertex set S⊆V of size at least l 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 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 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|−l) search tree algorithm [Schafer et al., Optim. Lett. 2012] cannot be improved and that, unless NP⊆coNP/poly, there is no polynomial kernel for the dual parameter |V|−l. Finally, we show that, in spite of this lower bound, the search tree algorithm for the dual parameter |V|−l can be tuned into an efficient exact algorithm for 2-Club that substantially outperforms previous implementations.