p-Edge/Vertex-Connected Vertex Cover: Parameterized and Approximation Algorithms
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We introduce and study two natural generalizations of the Connected VertexCover (VC) problem: the p-Edge-Connected and p-Vertex-Connected VC problem (where p ≥ 2 is a fixed integer). Like Connected VC, both new VC problems are FPT, but do not admit a polynomial kernel unless NP ⊆ coNP/poly, which is highly unlikely. We prove however that both problems admit time efficient polynomial sized approximate kernelization schemes. We obtain an O(2^O(pk)n^O(1))-time algorithm for the p-Edge-Connected VC and an O(2^O(k^2)n^O(1))-time algorithm for the p-Vertex-Connected VC. Finally, we describe a 2(p+1)-approximation algorithm for the p-Edge-Connected VC. The proofs for the new VC problems require more sophisticated arguments than for Connected VC. In particular, for the approximation algorithm we use Gomory-Hu trees and for the approximate kernels a result on small-size spanning p-vertex/edge-connected subgraph of a p-vertex/edge-connected graph obtained independently by Nishizeki and Poljak (1994) and Nagamochi and Ibaraki (1992).
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