Skip to main content
SpringerLink
Log in

Approximating the two-level facility location problem via a quasi-greedy approach

  • Published:
Mathematical Programming Submit manuscript

Abstract

We propose a quasi-greedy algorithm for approximating the classical uncapacitated 2-level facility location problem (2-LFLP). Our algorithm, unlike the standard greedy algorithm, selects a sub-optimal candidate at each step. It also relates the minimization 2-LFLP problem, in an interesting way, to the maximization version of the single level facility location problem. Another feature of our algorithm is that it combines the technique of randomized rounding with that of dual fitting.

This new approach enables us to approximate the metric 2-LFLP in polynomial time with a ratio of 1.77, a significant improvement on the previously known approximation ratios. Moreover, our approach results in a local improvement procedure for the 2-LFLP, which is useful in improving the approximation guarantees for several other multi-level facility location problems. An additional result of our approach is an O(ln (n))-approximation algorithm for the non-metric 2-LFLP, where n is the number of clients. This is the first non-trivial approximation for a non-metric multi-level facility location 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., Labbe, M., Leung, J., Queyranne, M.: On the two-level uncapacitated facility location problem. INFORMS Journal on Computing 8, 289–301 (1996)

    MATH  Google Scholar 

  2. Aardal, K., Chudak, F.A., Shmoys, D.B.: A 3-approximation algorithm for the k-level uncapacitated facility location problem. Information Processing Letters 72, 161–167 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  3. Ageev, A.: Improved approximation algorithms for multilevel facility location problems. Oper. Res. Letters 30, 327–332 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Ageev, A., Sviridenko, M.: An 0.828-approximation algorithm for uncapacitated facility location problem. Discrete Applied Mathematics 93, 289–296, (1999)

    Google Scholar 

  5. Ageev, A., Ye, Y., Zhang, J.: Improved Combinatorial Apporixmation Algorithms for the k-Level Facility Location Problem. In: Proceedings of The 30th International Colloquium on Automata, Languages and Programming (ICALP), LNCS 2719, 2003, pp. 145–156

  6. Bumb, A.: An approximation algorithm for the maximization version of the two level uncapacitated facility location problem. Operations Research Letters 29(4), 155–161 (2001)

    Article  MathSciNet  Google Scholar 

  7. Bumb, A.F., Kern, W.: A simple dual ascent algorithm for the multilevel facility location problem. In: 4th International Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX 2001), LNCS 2129, 2001, pp. 55–62

  8. Barros, A.I., Labbe, M.: A general model for the uncapacitated facility and depot location problem. Location Science 2, 173–191 (1994)

    MATH  Google Scholar 

  9. Carr, R.D., Fleischer, L., Leung, V.J., Phillips, C.A.: Strengthening integrality gaps for capacitated network design and covering problems. In: Proceedings of the 11th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2000, pp. 106–115

  10. Chardaire, P.: Facility Location Optimization and Cooperative Games. Ph.d. Thesis, School of Information Systems, University of East-Anglia, Norwich NR4 7TJ, UK, 1998

  11. Charikar, M., Chekuri, C., Cheung, T., Dai, Z., Goel, A., Guha, S., Li, M.: Approximation Algorithms for Directed Steiner Problems. Journal of Algorithms 33, 73–91 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Charikar, M., Guha, S.: Improved combinatorial algorithms for facility location problems. SIAM Journal on Computing 34(4), 803–824 (2005)

    Article  MathSciNet  Google Scholar 

  13. Chekuri, C., Even, G., Guy Kortsarz: A combinatorial approximation algorithm for the group Steiner problem. submitted to Discrete Applied Mathematics

  14. Cornuéjols, G., Fisher, M.L., Nemhauser, G.L.: Location of bank accounts to optimize float: an analytic study of exact and approximate algorithms. Mngt Sci. 23, 789–810 (1977)

    MATH  Google Scholar 

  15. Cornuéjols, G., Nemhauser, G.L., Wolsey, L.A.: The uncapacitated facility location problem. In: P. Mirchandani and R. Francis, (eds.) Discrete Location Theory, Wiley, New York, 1990, pp. 119–171

  16. Chudak, F.A., Shmoys, D.B.: Improved approximation algorithms for the uncapacitated facility location problem. SIAM Journal on Computing 33, 1–25 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Edwards, N.: Approximation algorithms for the multi-level facility location problem. Ph.D. Thesis, School of Operations Research and Industrial Engineering, Cornell University, 2001

  18. Erdös, P., Selfridge, J.L.: On a combinatorial game. J. Combinatorial Theory, Ser. A 14, 298–301 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  19. Feige, U.: A threshold of ln n for approximating set cover. Journal of the ACM 45, 634–652 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Gao, J.J., Robinson, E.P. Jr.: A dual-based optimization procedure for the two-echelon uncapacitated facility location problem. Naval Research Logistics 839, 191–212 (1992)

    Google Scholar 

  21. Gao, J.J., Robinson, E.P. Jr.: Uncapacitated facility location: General solution procedure and computational experience. European Journal of Operational Research 76, 410–427 (1994)

    Article  MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  23. Hochbaum, D.S.: Heuristics for the fixed cost median problem. Mathematical Programming 22(2), 148–162 (1982)

    Article  MathSciNet  Google Scholar 

  24. Kaufman, L., Eede, M.V., Hansen, P.: A plant and warehouse location problem. Operations Research Quarterly 28, 547–554 (1977)

    Article  MATH  Google Scholar 

  25. Labbe, M.: The multi-level uncapacitated facility location problem is not submodular. European Journal of Operational Research 72, 607–609 (1996)

    Google Scholar 

  26. Levin, A.: Personal Communication, 2003

  27. Guha, S., Meyerson, A., Munagala, K.: Hierarchical placement and network design problems. In: Proceedings of the 41st IEEE Symposium on Foundations of Computer Science (FOCS), 2000, pp. 603–612

  28. Jain, K., Mahdian, M., Saberi, A.: A new greedy approach for facility location problems. In: Proceedings of the 34th ACM Symposium on Theory of Computing (STOC), 2002, pp. 731–740

  29. Jain, K., Mahdian, M., Salavatipour, M.R.: Packing Steiner trees. In: Proceedings of the 14th ACM-SIAM Symposium on Discrete Algorithms (SODA), 2003, pp. 266–274

  30. Mahdian, M., Ye, Y., Zhang, J.: Improved approximation algorithms for metric facility location problems. In: 5th International Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX 2002), Lecture Notes in Computer Science, 2462, 2002, pp. 229–242

  31. Mahdian, M., Ye, Y., Zhang, J.: A 2-approximation algorithm for the soft-capacitated facility location problem. In: 6th International Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX 2003), Lecture Notes in Computer Science, 2764, 2003, pp. 129–140

  32. Meyerson, A., Munagala, K., Plotkin, S.: Cost-distance: two-metric network design. In: Proceedings of the 41st IEEE Symposium on Foundations of Computer Science (FOCS), 2000, pp. 624–630

  33. Ro, H., Tcha, D.: A branch-and-bound algorithm for the two-level uncapacitated facility location problem with some side constraints. European Journal of Operational Research 18, 349–358 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  34. Robinson, E.P. Jr., Gao, L.: A new formulation and linear programming based optimization procedure for the two-echelon uncapacitated facility location problem. Annals of the Society of Logistics Engineers 2, 39–59

  35. Robinson, E.P. Jr., Gao, L., Muggenborg, S.D.: Designing an integrated distribution system at DowBrands Inc. Interfaces 23, 107–117 (1993)

    Article  Google Scholar 

  36. Shmoys, D.B.: Approximation algorithms for facility location problems. In: 3rd International Workshop on Approximation Algorithms for Combinatorial Optimization (APPROX), LNCS 1913, 2000, pp. 27–33

  37. Shmoys, D.B., Tardos, E., Aardal, K.I.: Approximation algorithms for facility location problems. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing 29th STOC, 1997, pp. 265–274

  38. Tcha, D., Lee, B.: A branch-and-bound algorithm for the multi-level uncapacitated facility location problem. European Journal of Operational Research 18, 35–43 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  39. Zhang, J., Ye, Y.: A Note on the Maximization Version of the Multi-level Facility Location Problem. Operations Research Letters 30(5), 333–335 (2002)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IOMS-Operations Management, Stern School of Business, New York University, 44 W. 4th Street, Suite 8-66, New York, NY, 10012-1126, USA

    Jiawei Zhang

Authors
  1. Jiawei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiawei Zhang.

Additional information

An extended abstract of this paper appeared in the Proceedings of the 15th ACM-SIAM Symposium on Discrete Algorithms (SODA), January 2004.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Zhang, J. Approximating the two-level facility location problem via a quasi-greedy approach. Math. Program. 108, 159–176 (2006). https://doi.org/10.1007/s10107-006-0704-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-006-0704-x

Keywords

Search

Navigation