Primal-Dual Algorithms for Connected Facility Location Problems

  • Chaitanya Swamy
  • Amit Kumar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2462)


We consider the Connected Facility Location problem. We are given a graph G = (V,E) with cost c e on edge e, a set of facilities FV, and a set of demands DV. We are also given a parameter M ≥ 1. A solution opens some facilities, say F, assigns each demand j to an open facility i(j), and connects the open facilities by a Steiner tree T. The cost incurred is \( \sum {_{i \in F} f_i } + \sum {_{j \in \mathcal{D}} d_j c_{i\left( j \right)j} } + M\sum {_{e \in T} c_e } \) . We want a solution of minimum cost. A special case is when all opening costs are 0 and facilities may be opened anywhere, i.e.,F = V. If we know a facility v that is open, then this problem reduces to the rent-or-buy problem. We give the first primal-dual algorithms for these problems and achieve the best known approximation guarantees. We give a 9-approximation algorithm for connected facility location and a 5-approximation for the rent-or-buy problem. Our algorithm integrates the primal-dual approaches for facility location [7] and Steiner trees [1],[2]. We also consider the connected k-median problem and give a constant-factor approximation by using our primal-dual algorithm for connected facility location. We generalize our results to an edge capacitated version of these problems.


Facility Location Steiner Tree Facility Location Problem Demand Point 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Agrawal, P. Klein, and R. Ravi. When trees collide: An approximation algorithm for the generalized Steiner problem on networks. SIAM Journal on Computing, 24(3):440–456, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    M. X. Goemans and D. P. Williamson. A general approximation technique for constrained forest problems. SIAM Journal on Computing, 24:296–317, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    M. X. Goemans and D. P. Williamson. The primal-dual method for approximation algorithms and its application to network design problems. In D. S. Hochbaum, editor, Approximation Algorithms for NP-Hard Problems, chapter 4, pages 144–191. PWS Publishing Company, 1997.Google Scholar
  4. 4.
    S. Guha and S. Khuller. Connected facility location problems. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 40:179–190, 1997.MathSciNetGoogle Scholar
  5. 5.
    A. Gupta, J. Kleinberg, A. Kumar, R. Rastogi, and B. Yener. Provisioning a virtual private network: A network design problem for multicommodity flow. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC), pages 389–398, 2001.Google Scholar
  6. 6.
    R. Hassin, R. Ravi, and F. S. Selman. Approximation algorithms for a capacitated network design problem. In Proceedings of 4th APPROX, pages 167–176, 2000.Google Scholar
  7. 7.
    K. Jain and V. V. Vazirani. Primal-dual approximation algorithms for metric facility location and k-median problems. Journal of the ACM, 48:274–296, 2001.CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    D. R. Karger and M. Minko.. Building Steiner trees with incomplete global knowledge. In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 613–623, 2000.Google Scholar
  9. 9.
    S. Khuller and A. Zhu. The general Steiner tree-star problem. Information Processing Letters, 2002. To appear.Google Scholar
  10. 10.
    Tae Ung Kim, Timothy J. Lowe, Arie Tamir, and James E. Ward. On the location of a tree-shaped facility. Networks, 28(3):167–175, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    A. Kumar, A. Gupta, and T. Roughgarden. A constant-factor approximation algorithm for the multicommodity rent-or-buy problem. In Proceedings of the 43 rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), 2002. To appear.Google Scholar
  12. 12.
    M. Labbé, G. Laporte, I. Rodrígues Martin, and J. J. Salazar González. The median cycle problem. Technical Report 2001/12, Department of Operations Research and Multicriteria Decision Aid at Université Libre de Bruxelles, 2001.Google Scholar
  13. 13.
    Y. Lee, S. Y. Chiu, and J. Ryan. A branch and cut algorithm for a Steiner tree-star problem. INFORMS Journal on Computing, 8(3):194–201, 1996.zbMATHCrossRefGoogle Scholar
  14. 14.
    P. Mirchandani and R. Francis, eds. Discrete Location Theory. John Wiley and Sons, Inc., New York, 1990.zbMATHGoogle Scholar
  15. 15.
    R. Ravi and A. Sinha. Integrated logistics: Approximation algorithms combining facility location and network design. In Proceedings of 9th IPCO, pages 212–229, 2002.Google Scholar
  16. 16.
    G. Robins and A. Zelikovsky. Improved steiner tree approximation in graphs. In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 770–779, 2000.Google Scholar
  17. 17.
    D. P. Williamson. The primal-dual method for approximation algorithms. Mathematical Programming, Series B, 91(3):447–478, 2002.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Chaitanya Swamy
    • 1
  • Amit Kumar
    • 1
  1. 1.Department of Computer ScienceCornell UniversityIthacaUSA

Personalised recommendations