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Integrated Logistics: Approximation Algorithms Combining Facility Location and Network Design

  • Ramamoorthi Ravi
  • Amitabh Sinha
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2337)

Abstract

We initiate a study of the approximability of integrated logistics problems that combine elements of facility location and the associated transport network design.

In the simplest version, we are given a graph G = (V, E) with metric edge costs c, a set of potential facilities Open image in new window with nonnegative facility opening costs φ, a set of clients D ⊆V (each with unit demand), and a positive integer u (cable capacity). We wish to open facilities and construct a network of cables, such that every client is served by some open facility and all cable capacities are obeyed. The objective is to minimize the sum of facility opening and cable installation costs. With only one zero-cost facility and infinite u, this is the Steiner tree problem, while with unit capacity cables this is the Uncapacitated Facility Location problem. We give a (ρstufl)-approximation algorithm for this problem, where ρp denotes any approximation ratio for problem P.

For an extension when the facilities don’t have costs but no more than p facilities may be opened, we provide a bicriteria approximation algorithm that has total cost at most ρp -median + 2 times the minimum but opens up to 2p facilities.

Finally, for the general version with k different types of cables, we extend the techniques of [Guha, Meyerson, Munagala, STOC 2001] to provide an O(k) approximation.

Keywords

Approximation Algorithm Facility Location Steiner Tree Facility Location Problem Network Design Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Agrawal, A., Klein, P., Ravi, R.: When trees collide: An approximation algorithm for the generalized Steiner problem on networks. SIAM J. Computing, 24:440–456, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Andrews, M., Zhang, L.: The access network design problem. Proc. of the 39th Ann. IEEE Symp. on Foundations of Computer Science, pages 42–49, 1998.Google Scholar
  3. 3.
    Arya, V., Garg, N., Khandekar, R., Pandit, V., Meyerson, A., Munagala, K.: Local search heuristic for k-median and facility location problems. Proc. 33rd ACM Symposium on the Theory of Computing, pages 21–29, 2001.Google Scholar
  4. 4.
    Awerbuch, B., Azar, Y.: Buy at bulk network design. Proc. 38th Ann. IEEE Symposium on Foundations of Computer Science, pages 542–547, 1997.Google Scholar
  5. 5.
    Balinski, M.L.: On finding integer solutions to linear programs. Proc. IBM Scientific Computing Symposium on Combinatorial Problems, pages 225–248, 1966.Google Scholar
  6. 6.
    Charikar, M., Guha, S.: Improved combinatorial algorithms for the facility location and k-median problems. Proc. 40th Ann. IEEE Symposium on Foundations of Computer Science, pages 378–388, 1999.Google Scholar
  7. 7.
    Charikar, M., Guha, S., Shmoys, D., Tardos, E.: A constant-factor approximation algorithm for the k-median problem. Proc. 31st ACM Symposium on Theory of Computing, pages 1–10, 1999.Google Scholar
  8. 8.
    Garg, N., Khandekar, R., Konjevod, G., Ravi, R., Salman, F.S., Sinha, A.: On the integrality gap of a natural formulation of the single-sink buy-at-bulk network design problem. Proc. 8th Conference on Integer Programming and Combinatorial Optimization, pages 170–184, 2001.Google Scholar
  9. 9.
    Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. Proc. 9th ACM-SIAM Symposium on Discrete Algorithms, pages 649–657, 1998.Google Scholar
  10. 10.
    Guha, S., Meyerson, A., Munagala, K.: Hierarchical placement and network design problems. Proc. 41st Ann. IEEE Symposium on Foundations of Computer Science, pages 603–612, 2000.Google Scholar
  11. 11.
    Guha, S., Meyerson, A., Munagala, K.: A constant factor approximation for the single sink edge installation problems. Proc. 33rd ACM Symposium on the Theory of Computing, pages 383–388, 2001.Google Scholar
  12. 12.
    Hassin, R., Ravi, R., F.S. Salman, F.S.: Approximation algorithms for a capacitated network design problem. Approximation Algorithms for Combinatorial Optimization, Springer-Verlag Lecture Notes in Computer Science 1913, pages 167–176, 2000.CrossRefGoogle Scholar
  13. 13.
    Jain, K., Mahdian, M., Saberi, A.: A new greedy approach for facility location problems. To appear in Proc. 34th ACM Symposium on the Theory of Computing, 2002.Google Scholar
  14. 14.
    Jain, K., Vazirani, V.V.: Primal-dual approximation algorithms for metric facility location and k-median problems. Proc. 40th Ann. IEEE Symposium on Foundations of Computer Science, pages 2–13, 1999.Google Scholar
  15. 15.
    Karger, D., Minkoff, M.: Building Steiner trees with incomplete global knowledge. Proc. 41st Ann. IEEE Symposium on Foundations of Computer Science, pages 613–623, 2000.Google Scholar
  16. 16.
    Korupulu, M., Plaxton, C., Rajaraman, R.: Analysis of a local search heuristic for facility location problems. Proc. 9th ACM-SIAM Symposium on Discrete Algorithms, pages 1–10, 1998.Google Scholar
  17. 17.
    Mahdian, M., Ye, Y., Zhang, J.: A 1.52 approximation algorithm for the uncapacitated facility location problem. Manuscript, 2001.Google Scholar
  18. 18.
    Ravi, R.: A primal-dual approximation algorithm for the Steiner Forest Problem. Information Processing Letters 50(4): 185–190, 1994.MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Robins, G., Zelikovsky, A.: Improved Steiner tree approximation in graphs. Proc. 10th ACM-SIAM Symposium on Discrete Algorithms, pages 770–779, 1999.Google Scholar
  20. 20.
    Salman, F.S., Cheriyan, J., Ravi, R., Subramanian, S.: Buy-at-bulk network design: Approximating the single-sink edge installation problem. SIAM Journal on Optimization, 11:3, 595–610, 2000.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Shmoys, D., Tardos, E., Aardal, K.: Approximation algorithms for the facility location problem. Proc. 29th ACM Symposium on the Theory of Computing, pages 265–274, 1997.Google Scholar
  22. 22.
    Sviridenko, M.: A 1.582 approximation algorithm for the metric uncapacitated facility location problem. Proc. 9th Conference on Integer Programming and Combinatorial Optimization (2002), LNCS 2337, Springer Verlag, this volume.Google Scholar
  23. 23.
    Talwar, K.: Single sink buy-at-bulk LP has constant integrality gap. Proc. 9th Conference on Integer Programming and Combinatorial Optimization (2002), LNCS 2337, Springer Verlag, this volume.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Ramamoorthi Ravi
    • 1
  • Amitabh Sinha
    • 1
  1. 1.GSIACarnegie Mellon UniversityPittsburghUSA

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