Skip to main content

Approximation algorithms for the robust facility leasing problem

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

In this paper, we consider the robust facility leasing problem (RFLE), which is a variant of the well-known facility leasing problem. In this problem, we are given a facility location set, a client location set of cardinality n, time periods \(\{1, 2, \ldots , T\}\) and a nonnegative integer \(q < n\). At each time period t, a subset of clients \(D_{t}\) arrives. There are K lease types for all facilities. Leasing a facility i of a type k at any time period s incurs a leasing cost \(f_i^{k}\) such that facility i is opened at time period s with a lease length \(l_k\). Each client in \(D_t\) can only be assigned to a facility whose open interval contains t. Assigning a client j to a facility i incurs a serving cost \(c_{ij}\). We want to lease some facilities to serve at least \(n-q\) clients such that the total cost including leasing and serving cost is minimized. Using the standard primal–dual technique, we present a 6-approximation algorithm for the RFLE. We further offer a refined 3-approximation algorithm by modifying the phase of constructing an integer primal feasible solution with a careful recognition on the leasing facilities.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Anthony, B.M., Gupta, A.: Infrastructure leasing problems. In: Proceedings of International Conference on Integer Programming and Combinatorial Optimization, pp. 424–438. Springer, Berlin (2007)

  2. Arya, V., Garg, N., Khandekar, R., Meyerson, A., Munagala, K., Pandit, V.: Local search heuristics for \(k\)-median and facility location problems. SIAM J. Comput. 33, 544–562 (2004)

    Article  MathSciNet  MATH  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 problems. SIAM J. Comput. 34, 803–824 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Charikar, M., Khuller, S., Mount, DM., Narasimhan, G.: Algorithms for facility location problems with outliers. In: Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 642–651. Society for Industrial and Applied Mathematics (2001)

  6. 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 

  7. de Lima, M.S., San Felice, M.C., Lee, O.: Facility leasing with penalties. ArXiv:1610.00575 (2016)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  10. 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 

  11. Kling, P., auf der Heide, F.M., Pietrzyk, P.: An algorithm for online facility leasing. In: Proceedings of the 19th International Colloquium on Structural Information and Communication Complexity, pp. 61–72 (2012)

  12. Kuehn, A.A., Hamburger, M.J.: A heuristic program for locating warehouses. Manag. Sci. 9, 643–666 (1963)

    Article  Google Scholar 

  13. Li, S.: A 1.488 approximation algorithm for the uncapacitated facility location problem. Inf. Comput. 222, 45–58 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  15. Mahdian, M., Ye, Y., Zhang, J.: Improved approximation algorithms for metric facility location problems. In: Proceedings of International Workshop on Approximation Algorithms for Combinatorial Optimization, pp. 229–242. Springer, Berlin (2002)

  16. Manne, A.S.: Plant location under economies-of-scale-decentralization and computation. Manag. Sci. 11, 213–235 (1964)

    Article  Google Scholar 

  17. Nagarajan, C., Williamson, D.P.: Offline and online facility leasing. Discrete Optim. 10, 361–370 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. 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, pp. 265–274 (1997)

  19. Shu, J., Teo, C.P., Shen, Z.J.M.: Stochastic transportation-inventory network design problem. Oper. Res. 53, 48–60 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sviridenko, M.: An improved approximation algorithm for the metric uncapacitated facility location problem. In: Proceedings of International Conference on Integer Programming and Combinatorial Optimization, pp. 240–257. Springer, Berlin (2002)

  21. Teo, C.P., Shu, J.: Warehouse-retailer network design problem. Oper. Res. 52, 396–408 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The second author is supported by Natural Science Foundation of China (No. 11531014). The third author is supported by the Project Sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, and by Higher Educational Science and Technology Program of Shandong Province (No. J17KA171). The fourth author is supported by Higher Educational Science and Technology Program of Shandong Province (No. J15LN22) and the Science and Technology Development Plan Project of Jinan City (No. 201401211).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Dongmei Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Han, L., Xu, D., Li, M. et al. Approximation algorithms for the robust facility leasing problem. Optim Lett 12, 625–637 (2018). https://doi.org/10.1007/s11590-018-1238-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-018-1238-x

Keywords