2012 | OriginalPaper | Chapter
Parameterized Algorithmics and Computational Experiments for Finding 2-Clubs
Authors : Sepp Hartung, Christian Komusiewicz, André Nichterlein
Published in: Parameterized and Exact Computation
Publisher: Springer Berlin Heidelberg
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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 [Schäfer 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.