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Parameterized Complexity of Secluded Connectivity Problems

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Abstract

The Secluded Path problem models a situation where sensitive information has to be transmitted between a pair of nodes along a path in a network. The measure of the quality of a selected path is its exposure cost, which is the total cost of vertices in its closed neighborhood. The task is to select a secluded path, i.e., a path with a small exposure cost. Similarly, the Secluded Steiner Tree problem is to find a tree in a graph connecting a given set of terminals such that the exposure cost of the tree is minimized. In this paper we present a systematic study of the parameterized complexity of Secluded Steiner Tree. In particular, we establish the tractability of Secluded Path being parameterized by “above guarantee” value, which in this case is the length of a shortest path between vertices. We also show how to extend this result for Secluded Steiner Tree, in this case we parameterize above the size of an optimal Steiner tree and the number of terminals. We also consider various parameterization of the problems such as by the treewidth, the size of a vertex cover, feedback vertex set, or the maximum vertex degree and establish kernelization complexity of the problem subject to different choices of parameters.

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Notes

  1. Gutin et al. prove in [21] the statement for the dual Hitting Set problem.

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Correspondence to Petr A. Golovach.

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A preliminary version of this paper appeared as an extended abstract in the proceedings of FSTTCS 2015 [15]. The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n. 267959 and the Government of the Russian Federation (grant 14.Z50.31.0030).

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Fomin, F.V., Golovach, P.A., Karpov, N. et al. Parameterized Complexity of Secluded Connectivity Problems. Theory Comput Syst 61, 795–819 (2017). https://doi.org/10.1007/s00224-016-9717-x

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