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2019 | OriginalPaper | Buchkapitel

Algorithms and Complexity Results for the Capacitated Vertex Cover Problem

verfasst von : Sebastiaan B. van Rooij, Johan M. M. van Rooij

Erschienen in: SOFSEM 2019: Theory and Practice of Computer Science

Verlag: Springer International Publishing

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Abstract

We study the capacitated vertex cover problem (CVC). In this natural extension to the vertex cover problem, each vertex has a predefined capacity which indicates the total amount of edges that it can cover. In this paper, we study the complexity of the CVC problem. We give NP-completeness proofs for the problem on modular graphs, tree-convex graphs, and planar bipartite graphs of maximum degree three. For the first two graph classes, we prove that no subexponential-time algorithm exist for CVC unless the ETH fails.

Furthermore, we introduce a series of exact exponential-time algorithms which solve the CVC problem on several graph classes in \(\mathcal {O}((2 - \epsilon )^n)\) time, for some \(\epsilon > 0\). Amongst these graph classes are, graphs of maximum degree three, other degree-bounded graphs, regular graphs, graphs with large matchings, c-sparse graphs, and c-dense graphs. To obtain these results, we introduce an FPT treewidth algorithm which runs in \(\mathcal {O}^*((k + 1)^{tw})\) or \(\mathcal {O}^*(k^k)\) time, where k is the solution size and tw the treewidth, improving an earlier algorithm from Dom et al.

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Note that Vertex Cover equals Independet Set and Clique when considered from the viewpoint of exact exponential-time algorithms.

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Titel: Algorithms and Complexity Results for the Capacitated Vertex Cover Problem
verfasst von: Sebastiaan B. van Rooij
Johan M. M. van Rooij
Verlag: Springer International Publishing
Buch: SOFSEM 2019: Theory and Practice of Computer Science
Print ISBN: 978-3-030-10800-7

Electronic ISBN: 978-3-030-10801-4

Copyright-Jahr: 2019
DOI: https://doi.org/10.1007/978-3-030-10801-4_37

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