Skip to main content
SpringerLink
Log in

A primal-dual approximation algorithm for the k-prize-collecting minimum vertex cover problem with submodular penalties

  • Research Article
  • Published:
Frontiers of Computer Science Aims and scope Submit manuscript

Abstract

In this paper, we consider the k-prize-collecting minimum vertex cover problem with submodular penalties, which generalizes the well-known minimum vertex cover problem, minimum partial vertex cover problem and minimum vertex cover problem with submodular penalties. We are given a cost graph G = (V, E; c) and an integer k. This problem determines a vertex set S ⊆ V such that S covers at least k edges. The objective is to minimize the total cost of the vertices in S plus the penalty of the uncovered edge set, where the penalty is determined by a submodular function. We design a two-phase combinatorial algorithm based on the guessing technique and the primal-dual framework to address the problem. When the submodular penalty cost function is normalized and nondecreasing, the proposed algorithm has an approximation factor of 3. When the submodular penalty cost function is linear, the approximation factor of the proposed algorithm is reduced to 2, which is the best factor if the unique game conjecture holds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Karp R M. Reducibility among combinatorial problems. In: Miller R E, Thatcher J W, Bohlinger J D, eds. Complexity of Computer Computations. Boston: Springer, 1972, 85–103

    Chapter  Google Scholar 

  2. Vazirani V V. Approximation Algorithms. Berlin, Heidelberg: Springer, 2001

    MATH  Google Scholar 

  3. Khot S, Regev O. Vertex cover might be hard to approximate to within 2-ε. Journal of Computer and System Sciences, 2008, 74(3): 335–349

    Article  MathSciNet  Google Scholar 

  4. Hochbaum D S. Approximation algorithms for the set covering and vertex cover problems. SIAM Journal on Computing, 1982, 11(3): 555–556

    Article  MathSciNet  Google Scholar 

  5. Bar-Yehuda R, Even S. A linear-time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms, 1981, 2(2): 198–203

    Article  MathSciNet  Google Scholar 

  6. Bshouty N H, Burroughs L. Massaging a linear programming solution to give a 2-approximation for a generalization of the vertex cover problem. In: Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science. 1998, 298–308

  7. Hochbaum D S. The t-vertex cover problem: extending the half integrality framework with budget constraints. In: Proceedings of International Workshop on Approximation Algorithms for Combinatorial Optimization. 1998, 111–122

  8. Bar-Yehuda R. Using homogeneous weights for approximating the partial cover problem. Journal of Algorithms, 2001, 39(2): 137–144

    Article  MathSciNet  Google Scholar 

  9. Gandhi R, Khuller S, Srinivasan A. Approximation algorithms for partial covering problems. Journal of Algorithms, 2004, 53(1): 55–84

    Article  MathSciNet  Google Scholar 

  10. Mestre J. A primal-dual approximation algorithm for partial vertex cover: making educated guesses. Algorithmica, 2009, 55(1): 227–239

    Article  MathSciNet  Google Scholar 

  11. Hochbaum D S. Solving integer programs over monotone inequalities in three variables: a framework for half integrality and good approximations. European Journal of Operational Research, 2002, 140(2): 291–321

    Article  MathSciNet  Google Scholar 

  12. Bar-Yehuda R, Rawitz D. On the equivalence between the primal-dual schema and the local ratio technique. SIAM Journal on Discrete Mathematics, 2005, 19(3): 762–797

    Article  MathSciNet  Google Scholar 

  13. Li Y, Du D, Xiu N, Xu D. Improved approximation algorithms for the facility location problems with linear/submodular penalties. Algorithmica, 2015, 73(2): 460–482

    Article  MathSciNet  Google Scholar 

  14. Du D, Lu R, Xu D. A primal-dual approximation algorithm for the facility location problem with submodular penalties. Algorithmica, 2012, 63(1–2): 191–200

    Article  MathSciNet  Google Scholar 

  15. Liu X, Li W. Approximation algorithms for the multiprocessor scheduling with submodular penalties. Optimization Letters, 2021, 15(6): 2165–2180

    Article  MathSciNet  Google Scholar 

  16. Liu X, Li W, Xie R. A primal-dual approximation algorithm for the k-prize-collecting minimum power cover problem. Optimization Letters, 2021, DOI: https://doi.org/10.1007/s11590-021-01831-z

  17. Iwata S, Nagano K. Submodular function minimization under covering constraints. In: Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science. 2009, 671–680

  18. Xu D, Wang F, Du D, Wu C. Approximation algorithms for submodular vertex cover problems with linear/submodular penalties using primal-dual technique. Theoretical Computer Science, 2016, 630: 117–125

    Article  MathSciNet  Google Scholar 

  19. Kamiyama N. A note on the submodular vertex cover problem with submodular penalties. Theoretical Computer Science, 2017, 659: 95–97

    Article  MathSciNet  Google Scholar 

  20. Guo J S, Liu W, Hou B. An approximation algorithm for p-prize-collecting set cover problem. Journal of the Operations Research Society of China, 2021, DOI: https://doi.org/10.1007/s40305-021-00364-7

  21. Fleischer L, Iwata S. A push-relabel framework for submodular function minimization and applications to parametric optimization. Discrete Applied Mathematics, 2003, 131(2): 311–322

    Article  MathSciNet  Google Scholar 

  22. Kao M J, Shiau J Y, Lin C C, Lee D T. Tight approximation for partial vertex cover with hard capacities. Theoretical Computer Science, 2019, 778: 61–72

    Article  MathSciNet  Google Scholar 

  23. Cheung W C, Goemans M X, Wong S C W. Improved algorithms for vertex cover with hard capacities on multigraphs and hypergraphs. In: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms. 2014, 1714–1726

  24. Wong S C W. Tight algorithms for vertex cover with hard capacities on multigraphs and hypergraphs. In: Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms. 2017, 2626–2637

Download references

Acknowledgements

The work was supported in part by the National Natural Science Foundation of China (Grant No. 12071417).

Author information

Authors and Affiliations

  1. School of Information Science and Engineering, Yunnan University, Kunming, 650500, China

    Xiaofei Liu

  2. School of Mathematics and Statistics, Yunnan University, Kunming, 650500, China

    Weidong Li

  3. Dianchi College of Yunnan University, Kunming, 650228, China

    Jinhua Yang

Authors
  1. Xiaofei Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weidong Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jinhua Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Weidong Li.

Additional information

Xiaofei Liu received PhD degree in Mathematics from Yunnan University, China in 2018. He was a postdoctoral researcher at Peking University, China. He is currently a lecturer with Yunnan University, China. His research interests include theoretical computer science and discrete optimization.

Weidong Li received the PhD degree in Department of Mathematics, from Yunnan University, China in 2010. He is currently a professor at Yunnan University, China. His research interests include discrete optimization and algorithmic game theory.

Jinhua Yang is currently an associate professor at Dianchi College of Yunnan University, China. His research interests include image processing, deep learning and machine learning.

Electronic supplementary material

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, X., Li, W. & Yang, J. A primal-dual approximation algorithm for the k-prize-collecting minimum vertex cover problem with submodular penalties. Front. Comput. Sci. 17, 173404 (2023). https://doi.org/10.1007/s11704-022-1665-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11704-022-1665-9

Keywords

Search

Navigation