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On Colorful Vertex and Edge Cover Problems

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Abstract

In this paper, we study two generalizations of Vertex Cover and Edge Cover, namely Colorful Vertex Cover and Colorful Edge Cover. In the Colorful Vertex Cover problem, given an n-vertex edge-colored graph G with colors from \(\{1, \ldots , \omega \}\) and coverage requirements \(r_1, r_2, \ldots , r_\omega \), the goal is to find a minimum-sized set of vertices that are incident on at least \(r_i\) edges of color i, for each \(1 \le i \le \omega \), i.e., we need to cover at least \(r_i\) edges of color i. Colorful Edge Cover is similar to Colorful Vertex Cover, except here we are given a vertex-colored graph and the goal is to cover at least \(r_i\) vertices of color i, for each \(1 \le i \le \omega \), by a minimum-sized set of edges. These problems have several applications in fair covering and hitting of geometric set systems involving points and lines that are divided into multiple groups. Here, “fairness” ensures that the coverage (resp. hitting) requirement of every group is fully satisfied. We obtain a \((2+\epsilon )\)-approximation for the Colorful Vertex Cover problem in time \(n^{O(\omega /\epsilon )}\). Thus, for a constant number of colors, the problem admits a \((2+\epsilon )\)-approximation in polynomial time. Next, for the Colorful Edge Cover problem, we design an \(O(\omega n^3)\) time exact algorithm, via a chain of reductions to a matching problem. For all intermediate problems in this chain of reductions, we design polynomial-time algorithms, which might be of independent interest.

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Notes

  1. The term “fair” stresses on the fact, in an abstract manner, that the resources should be divided evenly among different groups.

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Acknowledgements

The authors would like to thank Tanmay Inamdar and Kasturi Varadarajan for helpful discussions. The authors are also thankful to the anonymous reviewers. The research of the first author is partly funded by the European Research Council (ERC) via grant LOPPRE, reference 819416.

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Authors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Sayan Bandyapadhyay

  2. School of Computer Sciences, NISER, Bhubaneswar, India

    Aritra Banik

  3. Indian Institute of Technology Bombay, Mumbai, India

    Sujoy Bhore

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  1. Sayan Bandyapadhyay
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  2. Aritra Banik
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  3. Sujoy Bhore
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Contributions

Sayan Bandyapadhyay: Conceptualization, Investigation, Formal analysis, Methodology, Writing - original draft.

Aritra Banik: Conceptualization, Investigation, Formal analysis, Methodology, Writing - original draft.

Sujoy Bhore: Conceptualization, Investigation, Formal analysis, Methodology, Writing - original draft.

Corresponding author

Correspondence to Sujoy Bhore.

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A preliminary version of this article appeared in the proceedings of the 47th International Workshop on Graph-Theoretic Concepts in Computer Science (WG) 2021 [2].

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Bandyapadhyay, S., Banik, A. & Bhore, S. On Colorful Vertex and Edge Cover Problems. Algorithmica 85, 3816–3827 (2023). https://doi.org/10.1007/s00453-023-01164-6

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