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Some Results on Approximability of Minimum Sum Vertex Cover

Author Aleksa Stanković

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Author Details

Aleksa Stanković
  • Department of Mathematics, KTH Royal Institute of Technology, Stockholm, Sweden

Funding

Research supported by the Approximability and Proof Complexity project funded by the Knut and Alice Wallenberg Foundation.

Acknowledgements

Author thanks Johan Håstad for fruitful discussion, as well as Hans Oude Groeniger and anonymous reviewers for useful comments which improved the presentation of this work.

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Aleksa Stanković. Some Results on Approximability of Minimum Sum Vertex Cover. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.50

Abstract

We study the Minimum Sum Vertex Cover problem, which asks for an ordering of vertices in a graph that minimizes the total cover time of edges. In particular, n vertices of the graph are visited according to an ordering, and for each edge this induces the first time it is covered. The goal of the problem is to find the ordering which minimizes the sum of the cover times over all edges in the graph.
In this work we give the first explicit hardness of approximation result for Minimum Sum Vertex Cover. In particular, assuming the Unique Games Conjecture, we show that the Minimum Sum Vertex Cover problem cannot be approximated within 1.014. The best approximation ratio for Minimum Sum Vertex Cover as of now is 16/9, due to a recent work of Bansal, Batra, Farhadi, and Tetali.
We also revisit an approximation algorithm for regular graphs outlined in the work of Feige, Lovász, and Tetali, and show that Minimum Sum Vertex Cover can be approximated within 1.225 on regular graphs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Approximation algorithms
  • Theory of computation → Approximation algorithms analysis
Keywords
  • Hardness of approximation
  • approximability
  • approximation algorithms
  • Label Cover
  • Unique Games Conjecture
  • Vertex Cover

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References

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