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DOI: 10.7155/jgaa.00352
Parameterized Algorithmics and Computational Experiments for Finding 2Clubs
Vol. 19, no. 1, pp. 155190, 2015. Regular paper.
Abstract Given an undirected graph G=(V,E) and an integer l ≥ 1, the NPhard 2CLUB 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 2CLUB. On the positive side, we give polynomialsize 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, 2CLUB does not admit a polynomialsize 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−l}·VE)time search tree algorithm [SchÃ¤fer et al., Optim. Lett. 2012] cannot be improved and that, unless NP ⊆ coNP/poly, there is no polynomialsize problem 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 2CLUB that outperforms previous implementations.

Submitted: July 2013.
Reviewed: May 2014.
Revised: August 2014.
Accepted: February 2015.
Final: February 2015.
Published: March 2015.
Communicated by
Petra Mutzel
