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Parameterized Algorithms for Minimum Sum Vertex Cover

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LATIN 2024: Theoretical Informatics (LATIN 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14579))

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Abstract

Minimum sum vertex cover of an n-vertex graph G is a bijection \(\phi : V(G) \rightarrow [n]\) that minimizes the cost \(\sum _{\{u,v\} \in E(G)} \min \{\phi (u), \phi (v) \}\). Finding a minimum sum vertex cover of a graph (the MSVC problem) is NP-hard. MSVC is studied well in the realm of approximation algorithms. The best-known approximation factor in polynomial time for the problem is 16/9 [Bansal, Batra, Farhadi, and Tetali, SODA 2021]. Recently, Stankovic [APPROX/RANDOM 2022] proved that achieving an approximation ratio better than 1.014 for MSVC is NP-hard, assuming the Unique Games Conjecture. We study the MSVC problem from the perspective of parameterized algorithms. The parameters we consider are the size of a minimum vertex cover and the size of a minimum clique modulator of the input graph. We obtain the following results.

  • MSVC can be solved in \(2^{2^{O(k)}} n^{O(1)}\) time, where k is the size of a minimum vertex cover.

  • MSVC can be solved in \(f(k)\cdot n^{O(1)}\) time for some computable function f, where k is the size of a minimum clique modulator.

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Notes

  1. 1.

    Proofs of results marked with \(\star \) are omitted due to paucity of space.

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Authors and Affiliations

  1. Indian Institute of Technology Hyderabad, Sangareddy, India

    Shubhada Aute

  2. School of Computing, University of Leeds, Leeds, UK

    Fahad Panolan

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  1. Shubhada Aute
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  2. Fahad Panolan
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Correspondence to Shubhada Aute .

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Editors and Affiliations

  1. DIM-CMM, Universidad de Chile, Santiago, Chile

    José A. Soto

  2. Technical University of Munich, Munich, Germany

    Andreas Wiese

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Aute, S., Panolan, F. (2024). Parameterized Algorithms for Minimum Sum Vertex Cover. In: Soto, J.A., Wiese, A. (eds) LATIN 2024: Theoretical Informatics. LATIN 2024. Lecture Notes in Computer Science, vol 14579. Springer, Cham. https://doi.org/10.1007/978-3-031-55601-2_13

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  • DOI: https://doi.org/10.1007/978-3-031-55601-2_13

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