Complexity and approximability of certain bicriteria location problems

  • S. O. Krumke
  • H. Noltemeier
  • S. S. Ravi
  • M. V. Marathe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1017)

Abstract

We investigate the complexity and approximability of some location problems when two distance values are specified for each pair of potential sites. These problems involve the selection of a specified number of facilities (i.e. a placement of a specified size) to minimize a function of one distance metric subject to a budget constraint on the other distance metric. Such problems arise in several application areas including statistical clustering, pattern recognition and load-balancing in distributed systems. We show that, in general, obtaining placements that are near-optimal with respect to the first distance metric is NP-hard even when we allow the budget constraint on the second distance metric to be violated by a constant factor. However, when both the distance metrics satisfy the triangle inequality, we present approximation algorithms that produce placements which are near-optimal with respect to the first distance metric while violating the budget constraint only by a small constant factor. We also present polynomial algorithms for these problems when the underlying graph is a tree.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AI+91]
    A. Aggarwal, H. Imai, N. Katoh, and S. Suri. Finding k points with Minimum Diameter and Related Problems. J. Algorithms, 12(1):38–56, March 1991.CrossRefGoogle Scholar
  2. [BP91]
    J. Bar-Ilan and D. Peleg. Approximation Algorithms for Selecting Network Centers. In Proc. 2nd Workshop on Algorithms and Data Structures (WADS), pages 343–354, Ottawa, Canada, August 1991. Springer Verlag, LNCS vol. 519.Google Scholar
  3. [AA+94]
    B. Awerbuch, Y. Azar, A. Blum, and S. Vempala, Improved approximation guarantees for minimum-weight k-trees and prize-collecting salesmen. Proceedings of the 27th Annual ACM Symposium on the Theory of Computing (STOC'95), pp. 277–376.Google Scholar
  4. [BCV95]
    A. Blum, P. Chalasani and S. Vempala, A Constant-Factor Approximation for the k-MST Problem in the Plane. Proceedings of the 27th Annual ACM Symposium on the Theory of Computing (STOC'95), pp. 294–302.Google Scholar
  5. [BS94]
    M. Bellare and M. Sudan. Improved Non-Approximability Results. in Proceedings of the 26th annual ACM Symposium on the Theory of Computing (STOC), May 1994.Google Scholar
  6. [DF85]
    M.E. Dyer and A.M. Frieze. A Simple Heuristic for the p-Center Problem. Operations Research Letters, 3(6):285–288, Feb. 1985.MathSciNetGoogle Scholar
  7. [EN89]
    E. Erkut and S. Neuman. Analytical Models for Locating Undesirable Facilities. European J. Operations Research, 40:275–291, 1989.Google Scholar
  8. [FG88]
    T. Feder and D. Greene. Optimal Algorithms for Approximate Clustering. In ACM Symposium on Theory of Computing (STOC), pages 434–444, 1988.Google Scholar
  9. [GJ79]
    M.R. Garey and D.S. Johnson. Computers and Intractability. W.H. Freeman, 1979.Google Scholar
  10. [Go85]
    T.F. Gonzalez. Clustering to Minimize the Maximum Intercluster Distance. Theoretical Computer Science, 38:293–306, 1985.CrossRefGoogle Scholar
  11. [GH94]
    N. Garg, D. Hochbaum, An O(log n) Approximation for the k-minimum spanning tree problem in the plane. Proceedings of the 26th Annual ACM Symposium on the Theory of Computing (STOC'95), pp. 294–302.Google Scholar
  12. [HM79]
    G.Y. Handler and P.B. Mirchandani. Location on Networks: Theory and Algorithms. MIT Press, Cambridge, MA, 1979.Google Scholar
  13. [HS86]
    D. S. Hochbaum and D. B. Shmoys. A Unified Approach to Approximation Algorithms for Bottleneck Problems. Journal of the ACM, 33(3):533–550, July 1986.Google Scholar
  14. [KN+95a]
    S.O. Krumke, H. Noltemeier, S.S. Ravi and M.V. Marathe, Compact Location Problems with Budget and Communication Constraints. In 1st International Conference on Computing and Combinatorics, X'ian, China, August 1995.Google Scholar
  15. [Lee82]
    D.T. Lee. On k-nearest neighbor Voronoi diagrams in the plane. IEEE Trans. Comput., C-31:478–487, 1982.Google Scholar
  16. [LV92]
    J.H. Lin and J. S. Vitter. ɛ-Approximations with Minimum Packing Constraint Violation. In ACM Symposium on Theory of Computing (STOC), pages 771–781, May 1992.Google Scholar
  17. [MR+95]
    M.V. Marathe, R. Ravi, R. Sundaram, S.S. Ravi, D.J. Rosenkrantz, and H.B. Hunt III. Bicriteria Network Design Problems. To appear in Proceedings of the 22nd International Colloquium on Automata Languages and Programming (ICALP), 1995.Google Scholar
  18. [MF90]
    P.B. Mirchandani and R.L. Francis. Discrete Location Theory. Wiley-Interscience, New York, NY, 1990.Google Scholar
  19. [PS85]
    F.P. Preparata and M.I. Shamos. Computational Geometry: An Introduction. Springer-Verlag Inc., New York, NY, 1985.Google Scholar
  20. [RKM+93]
    V. Radhakrishnan, S.O. Krumke, M.V. Marathe, D.J. Rosenkrantz, and S.S. Ravi. Compact Location Problems. In 13th Conference on the Foundations of Software Technology and Theoretical Computer Science (FST-TCS), volume 761 of LNCS, pages 238–247, December 1993.Google Scholar
  21. [RMR+93]
    R. Ravi, M.V. Marathe, S.S. Ravi, D.J. Rosenkrantz, and H.B. Hunt III. Many birds with one stone: Multi-objective approximation algorithms. In Proceedings of the 25th Annual ACM Symposium on the Theory of Computing (STOC), pages 438–447, 1993.Google Scholar
  22. [RRT91]
    S.S. Ravi, D.J. Rosenkrantz, and G.K. Tayi. Facility Dispersion Problems: Heuristics and Special Cases. In Proc. 2nd Workshop on Algorithms and Data Structures (WADS), pages 355–366, Ottawa, Canada, August 1991. Springer Verlag, LNCS vol. 519. (Journal version: Operations Research, 42(2):299–310, March–April 1994.)Google Scholar
  23. [RR+94]
    R. Ravi, R. Sundaram, M. V. Marathe, D. J. Rosenkrantz, and S. S. Ravi, Spanning trees short or small. Proceedings, Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, (1994), pp 546–555. (Journal version to appear in SIAM Journal on Discrete Mathematics.)Google Scholar
  24. [MR+95]
    M. V. Marathe, R. Ravi, R. Sundaram, S. S. Ravi, D. J. Rosenkrantz, and H.B. Hunt III. Bicriteria Network Design problems, In Proceedings of the 22nd International Colloquium on Automata, Languages and Programming (ICALP), pages 438–447, 1993.Google Scholar
  25. [ST93]
    D. B. Shmoys and E. Tardos. Scheduling unrelated parallel machines with costs. In Proceedings of the 4th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 438–447, 1993.Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • S. O. Krumke
    • 1
  • H. Noltemeier
    • 1
  • S. S. Ravi
    • 2
  • M. V. Marathe
    • 3
  1. 1.University of WürzburgWürzburgGermany
  2. 2.University at Albany-SUNYAlbanyUSA
  3. 3.Los Alamos Nat. Lab.Los AlamosUSA

Personalised recommendations