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Approximation Algorithms for the Traveling Purchaser Problem and Its Variants in Network Design

  • R. Ravi
  • F. S. Salman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1643)

Abstract

The traveling purchaser problem is a generalization of the traveling salesman problem with applications in a wide range of areas including network design and scheduling. The input consists of a set of markets and a set of products. Each market offers a price for each product and there is a cost associated with traveling from one market to another. The problem is to purchase all products by visiting a subset of the markets in a tour such that the total travel and purchase costs are minimized. This problem includes many well-known NP-hard problems such as uncapacitated facility location, set cover and group Steiner tree problems as its special cases.

We give an approximation algorithm with a poly-logarithmic worst-case ratio for the traveling purchaser problem with metric travel costs. For a special case of the problem that models the ring-star network design problem, we give a constantfactor approximation algorithm. Our algorithms are based on rounding LP relaxation solutions.

Keywords

Approximation Algorithm Travel Salesman Problem Travel Salesman Problem Travel Cost Steiner Tree 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. [AMO 93]
    R. K. Ahuja, T. L. Magnanti and J. B. Orlin, Network flows: Theory, Algorithms and Applications, Prentice Hall, Englewood Cliffs, NJ, 1993. 32Google Scholar
  2. [AS 97]
    S. Arora and M. Sudan, “Improved low degree testing and its applications,” Proc. 29th ACM Annual Symp. on Theory of Computing, 485–495, 1997. 30Google Scholar
  3. [AKR 95]
    A. Agrawal, P. Klein and R. Ravi, “When trees collide: An approximation algorithm for the generalized Steiner problem on networks,” SIAM J. Computing 24, 440–456, 1995. 38zbMATHCrossRefMathSciNetGoogle Scholar
  4. [CCGG 98]
    M. Charikar, C. Chekuri, A. Goel and S. Guha, “Rounding via tree: Deterministic approximation algorithms for group Steiner trees and k-median,” Proc. 30th ACM Annual Symp. on Theory of Computing, 114–123, 1998. 33Google Scholar
  5. [CS 94]
    J. R. Current and D. A. Schilling, “The median tour and maximal covering tour problems: Formulations and heuristics,” European Journal of Operational Research, 73, 114–126, 1994.zbMATHCrossRefGoogle Scholar
  6. [F 96]
    U. Feige, “A threshold of ln n for approximating set cover,” Proc. 28th ACM Annual Symp. on Theory of Computing, 314–318, 1996. 30Google Scholar
  7. [FGT 97]
    M. Fischetti, J. S. Gonzalez and P. Toth, “A branch-and-cut algorithm for the symmetric generalized traveling salesman problem,” Operations Research, 45, 378–394, 1997. 33zbMATHMathSciNetCrossRefGoogle Scholar
  8. [GJ 79]
    M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman, San Francisco, 1979. 30, 35zbMATHGoogle Scholar
  9. [GKR 97]
    N. Garg, G. Konjevod and R. Ravi, “A poly-logarithmic approximation algorithm for the group Steiner tree problem,” Proc. of the 9th Ann. ACM-SIAM Symposium on Discrete Algorithms, 253–259, 1998. 31, 33Google Scholar
  10. [GLD 81]
    B. Golden, L. Levy and R. Dahl, “Two generalizations of the traveling salesman problem,” OMEGA, 9, 439–455, 1981. 30CrossRefGoogle Scholar
  11. [GLS 88]
    Martin Grötschel and Laszlo Lovász and Alexander Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer-Verlag, 1988. 32Google Scholar
  12. [Kli 98]
    J. G. Klincewicz, “Hub location in backbone/tributary network design: a review,” To appear, Location Science, 1998. 34Google Scholar
  13. [LV 92]
    J. H. Lin and J. S. Vitter, “ε-approximations with minimum packing constraint violation,” In Proc. of the 24th Ann. ACM Symp. on Theory of Computing, 771–782, May 1992. 31Google Scholar
  14. [MRS+ 95]
    M. V. Marathe, R. Ravi, R. Sundaram, S. S. Ravi, D. J. Rosenkratz and H. Hunt, “Bicriteria network design problems,” J. Algorithms, 28, 142–171, 1998. 31zbMATHCrossRefMathSciNetGoogle Scholar
  15. [Ong 82]
    H. L. Ong, “Approximate algorithms for the traveling purchaser problem,” Operations Research Letters, 1, 201–205, 1982. 29, 30zbMATHCrossRefGoogle Scholar
  16. [PC 98]
    W. L. Pearn and R. C. Chien, “Improved solutions for the traveling purchaser problem,” Computers and Operations Research, 25, 879–885, 1998. 30zbMATHCrossRefGoogle Scholar
  17. [Ram 81]
    T. Ramesh, “Traveling purchaser problem,” OPSEARCH, 18, 87–91, 1981. 29Google Scholar
  18. [RS 97]
    R. Raz and S. Safra, “A sub-constant error-probability low-degree test, and a subconstant error-probability PCP characterization of NP,” Proc. 29th Annual ACM Symp. on Theory of Computing, 314–318, 1997. 30Google Scholar
  19. [SvO 97]
    K. N. Singh and D. L. van Oudheusden, “A branch and bound algorithm for the traveling purchaser problem,” European Journal of Operational Research, 97, 571–579, 1997. 30zbMATHCrossRefGoogle Scholar
  20. [V 90]
    S. Voss, “Designing special communication networks with the traveling purchaser problem,” Proceedings of the First ORSA Telecommunications Conference, 106–110, 1990. 34Google Scholar
  21. [V 96]
    S. Voss, “Dynamic tabu search strategies for the traveling purchaser problem,” Annals of Operations Research, 63, 253–275, 1996. 30zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • R. Ravi
    • 1
  • F. S. Salman
  1. 1.GSIA, Carnegie Mellon UniversityPittsburghUSA

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