p-Edge/vertex-connected vertex cover: Parameterized and approximation algorithms

We introduce and study two natural generalizations of the Connected Vertex Cover (VC) problem: the p-Edge-Connected and p-Vertex-Connected VC problem (where p≥2 is a fixed integer). We obtain an 2O(pk)nO(1)-time algorithm for p-Edge-Connected VC and an 2O(k2)nO(1)-time algorithm for p-Vertex-Connect...

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Veröffentlicht in:Journal of computer and system sciences Jg. 133; S. 23 - 40
Hauptverfasser: Einarson, Carl, Gutin, Gregory, Jansen, Bart M.P., Majumdar, Diptapriyo, Wahlström, Magnus
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Elsevier Inc 01.05.2023
Schlagworte:
Approximate kernels
Approximation algorithms
Connectivity
Kernels
Parameterized algorithms
Vertex cover
Kernels
Vertex cover
Connectivity
Approximation algorithms
Approximate kernels
Parameterized algorithms
ISSN:0022-0000, 1090-2724
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Zusammenfassung:We introduce and study two natural generalizations of the Connected Vertex Cover (VC) problem: the p-Edge-Connected and p-Vertex-Connected VC problem (where p≥2 is a fixed integer). We obtain an 2O(pk)nO(1)-time algorithm for p-Edge-Connected VC and an 2O(k2)nO(1)-time algorithm for p-Vertex-Connected VC. Thus, like Connected VC, both constrained VC problems are FPT. Furthermore, like Connected VC, neither problem admits 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. 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 subgraphs of a p-vertex/edge-connected graph by Nishizeki and Poljak (1994) [30] and Nagamochi and Ibaraki (1992) [27].
ISSN:0022-0000
1090-2724
DOI:10.1016/j.jcss.2022.11.002