Abstract
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-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|−l·|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|−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 outperforms previous implementations.
Highlights
Finding cohesive subnetworks in a network is an important task in graphbased data mining and social network analysis
The kernelization results for these rather large parameters are motivated by our negative results: We show that 2-Club does not admit a polynomial kernel with respect to the size of a vertex cover of the underlying graph, unless NP ⊆ coNP/poly
We prove that unless the Strong Exponential Time Hypothesis (SETH)2 fails, s-Club cannot be solved in O∗((2 − )k ) time for all > 0
Summary
Finding cohesive subnetworks in a network is an important task in graphbased data mining and social network analysis. The kernelization results for these rather large parameters are motivated by our negative results: We show that 2-Club does not admit a polynomial kernel with respect to the size of a vertex cover of the underlying graph, unless NP ⊆ coNP/poly. This excludes polynomial kernels for many prominent structural parameters such as “feedback vertex set size”, pathwidth, and treewidth. We prove that unless the Strong Exponential Time Hypothesis (SETH) fails, s-Club cannot be solved in O∗((2 − )k ) time for all > 0 This is evidence that the previous search tree algorithm [22] is optimal with respect to the parameter k. Due to the space restrictions, most of the proofs are deferred to a long version of this article
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