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Modifying networks to obtain low cost trees

  • S. O. Krumke
  • H. Noltemeier
  • M. V. Marathe
  • S. S. Ravi
  • K. U. Drangmeister
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1197)

Abstract

We consider the problem of reducing the edge lengths of a given network so that the modified network has a spanning tree of small total length. It is assumed that each edge e of the given network has an associated function c e that specifies the cost of shortening the edge by a given amount and that there is a budget B on the total reduction cost. The goal is to develop a reduction strategy satisfying the budget constraint so that the total length of a minimum spanning tree in the modified network is the smallest possible over all reduction strategies that obey the budget constraint.

We show that in general the problem of computing an optimal reduction strategy for modifying the network as above is NP-hard and present the first polynomial time approximation algorithms for the problem, where the cost functions c e are allowed to be taken from a broad class of functions. We also present improved approximation algorithms for the class of treewidth-bounded graphs when the cost functions are linear. Our results can be extended to obtain approximation algorithms for more general network design problems such as those considered in [9, 10].

Keywords

Location Theory Approximation Algorithms Parametric Search Computational Complexity NP-hardness 

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References

  1. 1.
    O. Berman, “Improving The Location of Minisum Facilities Through Network Modification,” Annals of Operations Research, 40(1992), pp. 1–16.Google Scholar
  2. 2.
    J. Cong, A. B. Kahng, G. Robins, M. Sarafzadeh and C. K. Wong, “Provably Good Performance Driven Global Routing,” IEEE Transactions on Computer Aided Design, 11(6), 1992, pp. 739–752.Google Scholar
  3. 3.
    J. P. Cohoon and L. J. Randall, “Critical Net Routing,” IEEE Intern. Conf. on Computer Design, 1991, pp. 174–177.Google Scholar
  4. 4.
    T. H. Cormen, C. E. Leiserson and R. L. Rivest, Introduction to Algorithms, McGraw-Hill Book Co., Cambridge, MA, 1990.Google Scholar
  5. 5.
    W. Cunningham, “Optimal Attack and Reinforcement of a Network,” J. ACM, 32(3), 1985, pp. 549–561.Google Scholar
  6. 6.
    G.N. Frederickson and R. Solis-Oba, “Increasing the Weight of Minimum Spanning Trees”, Proceedings of the Sixth Annual ACM-SIAM SODA'96, Jan. 1996, pp. 539–546.Google Scholar
  7. 7.
    H. N. Gabow, Z. Galil, T. H. Spencer and R. E. Tarjan, “Efficient Algorithms for Finding Minimum Spanning Trees in Undirected and Directed Graphs,” Combinatorica, 6 (1986), pp. 109–122.Google Scholar
  8. 8.
    M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Co., San Francisco, CA, 1979.Google Scholar
  9. 9.
    M. X. Goemans, and D. P. Williamson, “A General Approximation Technique for Constrained Forest Problems”, SIAM J. Computing, 24 (1995), pp. 296–317.Google Scholar
  10. 10.
    M. X. Goemans, A. V. Goldberg, S. Plotkin, D. B. Shmoys, E. Tardos and D. P. Williamson, “Improved Approximation Algorithms for Network Design Problems,” Proceedings of the Fifth Annual ACM-SIAM SODA'94, Jan. 1994, pp. 223–232.Google Scholar
  11. 11.
    R. Hassin, “Approximation Schemes for the Restricted Shortest Path Problem,” Math. of OR, 17(1), 1992, pp. 36–42.Google Scholar
  12. 12.
    D. Karger and S. Plotkin, “Adding Multiple Cost Constraints to Combinatorial Optimization Problems, with Applications to Multicommodity Flows,” Proc. 27th Annual ACM Symp. on Theory of Computing (STOC'95), May 1995, pp. 18–25.Google Scholar
  13. 13.
    B. Kadaba and J. Jaffe, “Routing to Multiple Destinations in Computer Networks,” IEEE Trans. on Communication, Vol. COM-31, Mar. 1983, pp. 343–351.Google Scholar
  14. 14.
    V. P. Kompella, J. C. Pasquale and G. C. Polyzos, “Two Distributed Algorithms for the Constrained Steiner Tree Problem,” Technical Report CAL-1005-92, Computer Systems Laboratory, University of California, San Diego, Oct. 1992.Google Scholar
  15. 15.
    V. P. Kompella, J. C. Pasquale and G. C. Polyzos, “Multicast Routing for Multimedia Communication,” IEEE/ACM Transactions on Networking, 1993, pp. 286–292.Google Scholar
  16. 16.
    M. V. Marathe, R. Ravi, S. Sundaram, S. S. Ravi, D. J. Rosenkrantz, and H. B. Hunt III, “Bicriteria Network Design Problems”, Proc. ICALP'95, July 1995, pp. 487–498.Google Scholar
  17. 17.
    C. Phillips, “The Network Inhibition Problem,” Proc. 25th Annual ACM STOC'93, May 1993, pp. 288–293.Google Scholar
  18. 18.
    J. Plesnik, “The Complexity of Designing a Network with Minimum Diameter,” Networks, 11 (1981), pp. 77–85.Google Scholar
  19. 19.
    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,” Proc. 25th Annual ACM STOC'93, May 1993, pp. 438–447.Google Scholar
  20. 20.
    R. Ravi, “Rapid Rumor Ramification: Approximating the Minimum Broadcast Time,” Proceedings of the 35th Annual FOCS'94, Nov. 1994, pp. 202–213.Google Scholar
  21. 21.
    A. Warburton, “Approximation of Pareto optima in Multiple-Objective, Shortest Path Problems,” Oper. Res., 35 (1987), pp. 70–79.Google Scholar
  22. 22.
    Q. Zhu, M. Parsa and W. Dai, “An Iterative Approach for Delay Bounded Minimum Steiner Tree Construction,” Technical Report UCSC-CRL-94-39, University of California, Santa Cruz, Oct 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

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

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