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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.

References

  1. Andersson, A., Thorup, M.: Dynamic ordered sets with exponential search trees. J. ACM 54(3), 13 (2007)

    Article  MathSciNet  Google Scholar 

  2. Archetti, C., Feillet, D., Mor, A., Speranza, M.G.: Dynamic traveling salesman problem with stochastic release dates. Eur. J. Oper. Res. 280(3), 832–844 (2020)

    Article  MathSciNet  Google Scholar 

  3. Baswana, S., Gupta, M., Sen, S.: Fully dynamic maximal matching in \(O(\log n)\) update time. SIAM J. Comput. 44(1), 88–113 (2015)

    Article  MathSciNet  Google Scholar 

  4. Bhattacharya, S., Chakrabarty, D., Henzinger, M.: Fully dynamic approximate maximum matching and minimum vertex cover in \(O(\log ^3 n)\) worst case update time. In: Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms (SODA), Barcelona, Spain, pp. 470–489 (2017)

  5. Bhattacharya, S., Chakrabarty, D., Henzinger, M.: Deterministic fully dynamic approximate vertex cover and fractional matching in \(O(1)\) amortized update time. In: Proceedings of the 19th Conference on Integer Programming and Combinatorial Optimization (IPCO), Waterloo, Canada, pp. 86–98 (2017)

  6. Bhattacharya, S., Henzinger, M., Italiano, G.F.: Deterministic fully dynamic data structures for vertex cover and matching. In: Proceedings of the 26th ACM-SIAM Symposium on Discrete Algorithms (SODA), Philadelphia, USA, pp. 785–804 (2015)

  7. Bhattacharya, S., Henzinger, M., Italiano, G.F.: Design of dynamic algorithms via primal-dual method. In: Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming (ICALP), Heidelberg, Germany, vol. 2015, pp. 206–218 (2015)

  8. Buriol, L.S., Resende, M.G.C., Thorup, M.: Speeding up dynamic shortest-path algorithms. Inf. J. Comput. 20(2), 191–204 (2008)

    Article  MathSciNet  Google Scholar 

  9. Chuzhoy, J., Naor, J.: Covering problems with hard capacities. SIAM J. Comput. 36(2), 498–515 (2006)

    Article  MathSciNet  Google Scholar 

  10. Demetrescu, C., Italiano, G.F.: A new approach to dynamic all pairs shortest paths. J. ACM 51(6), 968–992 (2004)

    Article  MathSciNet  Google Scholar 

  11. Guha, S., Hassin, R., Khuller, S., Or, E.: Capacitated vertex covering. J. Algorithms 48(1), 257–270 (2003)

    Article  MathSciNet  Google Scholar 

  12. Gupta, A., Krishnaswamy, R., Kumar, A., Panigrahi, D.: Online and dynamic algorithms for set cover. In: Proceedings of the 49th ACM Symposium on Theory of Computing (STOC), Montreal, Canada, pp. 537–550 (2017)

  13. Holm, M.T.J., de Lichtenberg, K.: Poly-logarithmic deterministic fully-dynamic algorithms for connectivity, minimum spanning tree, 2-edge, and biconnectivity. J. ACM 48(4), 723–760 (2001)

    Article  MathSciNet  Google Scholar 

  14. Ivkovic, Z., Lloyd, E.L.: Fully dynamic maintenance of vertex cover. In: Proceedings of the 19th International Workshop on Graph-theoretic Concepts in Computer Science (WG), London, UK, pp. 99–111 (1994)

  15. Kejlberg-Rasmussen, C., Kopelowitz, T., Pettie, S., Thorup, M.: Faster worst case deterministic dynamic connectivity. In: Proceedings of the 24th Annual European Symposium on Algorithms (ESA), Aarhus, Denmark, vol. 57, pp. 1–15 (2016)

  16. Neiman, O., Solomon, S.: Simple deterministic algorithms for fully dynamic maximal matching. ACM Trans. Algorithms 12(1), 7 (2016)

    Article  MathSciNet  Google Scholar 

  17. Onak, K., Rubinfeld, R.: Maintaining a large matching and a small vertex cover. In: Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC), Cambridge, USA, pp. 457–464 (2010)

  18. Peleg, D., Solomon, S.: Dynamic (1+\(\epsilon \))-approximate matchings: a density-sensitive approach. In: Proceedings of the 27th ACM-SIAM Symposium on Discrete Algorithms (SODA), Virginia, USA, pp. 712–729 (2016)

  19. Silva, A., Aloise, D., Coelho, L.C., Rocha, C.: Heuristics for the dynamic facility location problem with modular capacities. Eur. J. Oper. Res. 290, 435–452 (2020)

    Article  MathSciNet  Google Scholar 

  20. Solomon, S.: Fully dynamic maximal matching in constant update time. In: Proceedings of the 57th Symposium on Foundations of Computer Science (FOCS), New Jersey, USA, pp. 325–334 (2016)

  21. Thorup, M.: Worst-case update times for fully-dynamic all-pairs shortest paths. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing (STOC), Baltimore, MD, USA, pp. 112–119 (2005)

  22. Yu, G., Yang, Y.: Dynamic routing with real-time traffic information. Oper. Res. 19, 1–26 (2017)

    Google Scholar 

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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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  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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