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Algorithms and Complexity Results for the Capacitated Vertex Cover Problem

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SOFSEM 2019: Theory and Practice of Computer Science (SOFSEM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11376))

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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Notes

  1. 1.

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

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Authors and Affiliations

  1. Department of Information and Computing Sciences, Utrecht University, PO Box 80.089, 3508 TB, Utrecht, The Netherlands

    Sebastiaan B. van Rooij & Johan M. M. van Rooij

Authors
  1. Sebastiaan B. van Rooij
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  2. Johan M. M. van Rooij
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Correspondence to Johan M. M. van Rooij .

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Editors and Affiliations

  1. University of Genoa, Genoa, Italy

    Barbara Catania

  2. Comenius University, Bratislava, Slovakia

    Rastislav Královič

  3. Poznań University of Technology, Poznań, Poland

    Jerzy Nawrocki

  4. Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

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van Rooij, S.B., van Rooij, J.M.M. (2019). Algorithms and Complexity Results for the Capacitated Vertex Cover Problem. In: Catania, B., Královič, R., Nawrocki, J., Pighizzini, G. (eds) SOFSEM 2019: Theory and Practice of Computer Science. SOFSEM 2019. Lecture Notes in Computer Science(), vol 11376. Springer, Cham. https://doi.org/10.1007/978-3-030-10801-4_37

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  • DOI: https://doi.org/10.1007/978-3-030-10801-4_37

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