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Approximating Dynamic Weighted Vertex Cover with Soft Capacities

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Abstract

This study considers the soft capacitated vertex cover problem in a dynamic setting. This problem generalizes the dynamic model of the vertex cover problem, which has been intensively studied in recent years. Given a dynamically changing vertex-weighted graph \(G=(V,E)\), which allows edge insertions and edge deletions, the goal is to design a data structure that maintains an approximate minimum vertex cover while satisfying the capacity constraint of each vertex. That is, when picking a copy of a vertex v in the cover, the number of v’s incident edges covered by the copy is up to a given capacity of v. We extend Bhattacharya et al.’s work [SODA’15 and ICALP’15] to obtain a deterministic primal-dual algorithm for maintaining a constant-factor approximate minimum capacitated vertex cover with \(O(\log n / \epsilon )\) amortized update time, where n is the number of vertices in the graph. The algorithm can be extended to (1) a more general model in which each edge is associated with a non-uniform and unsplittable demand, and (2) the more general capacitated set cover problem.

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Notes

  1. We thank the original author of [4] for clarifying the dependency of \(\epsilon \) in the update time.

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Acknowledgements

The extended abstract of this paper has been published in Proceedings of the 21st International Conference on Approximation Algorithms for Combinatorial Optimization Problems (APPROX’2018). We also thank the support from National Center for High-performance Computing (NCHC) and the Brain Research Center under the Higher Education Sprout Project.

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Authors and Affiliations

  1. Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu, 30013, Taiwan

    Hao-Ting Wei & Chung-Shou Liao

  2. Department of Computer Science, National Tsing Hua University, Hsinchu, 30013, Taiwan

    Wing-Kai Hon

  3. Department of Mathematics, University of Denver, Denver, USA

    Paul Horn

  4. Department of Mathematical Informatics, The University of Tokyo, Tokyo, Japan

    Kunihiko Sadakane

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  1. Hao-Ting Wei
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  2. Wing-Kai Hon
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  3. Paul Horn
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  4. Chung-Shou Liao
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  5. Kunihiko Sadakane
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Correspondence to Chung-Shou Liao.

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This work was partially supported by NSA Young Investigator Grant H98230-15-1-0258, Simons Collaboration Grant #525039, and MOST Taiwan Grants 105-2628-E-007-010-MY3 and 109-2634-F-007-018.

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Wei, HT., Hon, WK., Horn, P. et al. Approximating Dynamic Weighted Vertex Cover with Soft Capacities. Algorithmica 84, 124–149 (2022). https://doi.org/10.1007/s00453-021-00886-9

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