Skip to main content
SpringerLink
Log in

A Primal-Dual Approximation Algorithm for the Facility Location Problem with Submodular Penalties

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We consider the facility location problem with submodular penalties (FLPSP), introduced by Hayrapetyan et al. (Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 933–942, 2005), who presented a 2.50-approximation algorithm that is non-combinatorial because this algorithm has to solve the LP-relaxation of an integer program with exponential number of variables. The only known polynomial algorithm for this exponential LP is via the ellipsoid algorithm as the corresponding separation problem for its dual program can be solved in polynomial time. By exploring the properties of the submodular function, we offer a primal-dual 3-approximation combinatorial algorithm for this problem.

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. Aardal, K.I., Chudak, F.A., Shmoys, D.B.: A 3-approximation algorithm for the k-level uncapacitated facility location problem. Inf. Process. Lett. 72, 161–167 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ageev, A., Ye, Y., Zhang, J.: Improved combinatorial approximation algorithms for the k-level facility location problem. SIAM J. Discrete Math. 18, 207–217 (2003)

    Article  MathSciNet  Google Scholar 

  3. Byrka, J., Aardal, K.I.: An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem. SIAM J. Comput. 39, 2212–2231 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  4. Charikar, M., Guha, S.: Improved combinatorial algorithms for facility location and k-median problems. In: Proceedings of the Fortieth Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 378–388 (1999)

    Google Scholar 

  5. Charikar, M., Khuller, S., Mount, D.M., Naraasimban, G.: Algorithms for facility location problems with outliers. In: Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 642–651 (2001)

    Google Scholar 

  6. Chudak, F.A., Nagano, K.: Efficient solutions to relaxations of combinatorial problems with submodular penalties via the Lovasz extension and non-smooth convex optimization. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 79–88 (2007)

    Google Scholar 

  7. Chudak, F.A., Shmoys, D.B.: Improved approximation algorithms for the uncapacitated facility location problem. SIAM J. Comput. 33, 1–25 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. Du, D., Wang, X., Xu, D.: An approximation algorithm for the k-level capacitated facility location problem. J. Comb. Optim. 20, 361–368 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. Guha, S., Khuller, S.: Greedy strikes back: improved facility location algorithms. J. Algorithms 31, 228–248 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hayrapetyan, A., Swamy, C., Tardos, E.: Network design for information networks. In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 933–942 (2005)

    Google Scholar 

  12. Jain, K., Vazirani, V.V.: Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation. J. ACM 48, 274–296 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jain, K., Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.V.: Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. J. ACM 50, 795–824 (2003)

    Article  MathSciNet  Google Scholar 

  14. Korupolu, M.R., Plaxton, C.G., Rajaraman, R.: Analysis of a local search heuristic for facility location problems. J. Algorithms 37, 146–188 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mahdian, M., Ye, Y., Zhang, J.: Approximation algorithms for metric facility location problems. SIAM J. Comput. 36, 411–432 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Shmoys, D.B., Tardos, E., Aardal, K.I.: Approximation algorithms for facility location problems (extended abstract). In: Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing (STOC), pp. 265–274 (1997)

    Chapter  Google Scholar 

  17. Sviridenko, M.: An improved approximation algorithm for the metric uncapacitated facility location problem. In: Proceedings of the Ninth Integer Programming and Combinatorial Optimization (IPCO), pp. 240–257 (2002)

    Google Scholar 

  18. Xu, D., Du, D.: The k-level facility location game. Oper. Res. Lett. 34, 421–426 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Xu, G., Xu, J.: An LP rounding algorithm for approximating uncapacitated facility location problem with penalties. Inf. Process. Lett. 94, 119–123 (2005)

    Article  MATH  Google Scholar 

  20. Xu, G., Xu, J.: An improved approximation algorithm for uncapacitated facility location problem with penalties. J. Comb. Optim. 17, 424–436 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Xu, D., Zhang, S.: Approximation algorithm for facility location with service installation costs. Oper. Res. Lett. 36, 46–50 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Zhang, J.: Approximating the two-level facility location problem via a quasi-greedy approach. Math. Program. 108, 159–176 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Zhang, J., Chen, B., Ye, Y.: A multiexchange local search algorithm for the capacitated facility location problem. Math. Oper. Res. 30, 389–403 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Business Administration, University of New Brunswick, Fredericton, NB, E3B 5A3, Canada

    Donglei Du

  2. Department of Applied Mathematics, Beijing University of Technology, 100 Pingleyuan, Chaoyang District, Beijing, 100124, P.R. China

    Ruixing Lu & Dachuan Xu

Authors
  1. Donglei Du
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ruixing Lu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dachuan Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dachuan Xu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Du, D., Lu, R. & Xu, D. A Primal-Dual Approximation Algorithm for the Facility Location Problem with Submodular Penalties. Algorithmica 63, 191–200 (2012). https://doi.org/10.1007/s00453-011-9526-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-011-9526-1

Keywords

Search

Navigation