Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs [chapter]

Sepp Hartung, Christian Komusiewicz, André Nichterlein
2012 Lecture Notes in Computer Science  
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
more » ... e 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.
doi:10.1007/978-3-642-33293-7_22 fatcat:5hbgpg6wuzbcbetfwpivstuxle