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A Subgradient Method to Improve Approximation Ratio in the Minimum Latency Problem

  • Bang Ban HaEmail author
  • Nghia Nguyen Duc
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 244)

Abstract

The Minimum Latency Problem (MLP) is a combinatorial optimization problem which has many practical applications. Recently, several approximation algorithms with guaranteed approximation ratio have been proposed to solve the MLP problem. These algorithms start with a set of solutions of the k −MST or k −troll problem, then convert the solutions into Eulerian tours, and finally, concatenate these Eulerian tours to obtain a MLP tour. In this paper, we propose an algorithm based on the principles of the subgradient method. It still uses the set of solutions of the k −MST or k −troll problem as an input, then modifies each solution into a tour with cost smaller than that of Eulerian tour and finally, uses obtained tours to construct a MLP tour. Since the low cost tours are used to build a MLP tour, we can expect the approximation ratio of obtained algorithm will be improved. In order to illustrate this intuition, we have evaluated the algorithm on five benchmark datasets. The experimental results show that approximation ratio of our algorithm is improved compared to the best well-known approximation algorithms.

Keywords

Approximation Ratio Real Dataset Benchmark Dataset Large Instance Small Instance 
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.
    Archer, A., Levin, A., Williamson, D.: A Faster, Better Approximation Algorithm for the Minimum Latency Problem. J. SIAM 37(1), 1472–1498 (2007)MathSciNetGoogle Scholar
  2. 2.
    Arora, S., Karakostas, G.: Approximation schemes for minimum latency problems. In: Proc. ACM STOC, pp. 688–693 (1999)Google Scholar
  3. 3.
    Ban, H.B., Nguyen, K., Ngo, M.C., Nguyen, D.N.: An efficient exact algorithm for Minimum Latency Problem. J. Progress of Informatics (10), 1–8 (2013)Google Scholar
  4. 4.
    Blum, A., Chalasani, P., Coppersmith, D., Pulleyblank, W., Raghavan, P., Sudan, M.: The minimum latency problem. In: Proc. ACM STOC, pp. 163–171 (1994)Google Scholar
  5. 5.
    Chaudhuri, K., Goldfrey, B., Rao, S., Talwar, K.: Path, Tree and minimum latency tour. In: Proc. IEEE FOCS, pp. 36–45 (2003)Google Scholar
  6. 6.
    Garg, N.: Saving an Epsilon: A 2-approximation for the k −MST Problem in Graphs. In: Proc. STOC, pp. 396–402 (2005)Google Scholar
  7. 7.
    Goemans, M., Kleinberg, J.: An improved approximation ratio for the minimum latency problem. In: Proc. ACM-SIAM SODA, pp. 152–158 (1996)Google Scholar
  8. 8.
    Held, M., Karp, R.M.: The travelling salesman problem and minimum spanning tree: part II. J. Mathematical Programming 1, 5–25 (1971)MathSciNetGoogle Scholar
  9. 9.
    Motzkin, T., Schoenberg, I.J.: The relaxation method for linear inequalities. J. Mathematics, 393–404 (1954)Google Scholar
  10. 10.
    Sahni, S., Gonzalez, T.: P-complete approximation problem. J. ACM 23(3), 555–565 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Salehipour, A., Sorensen, K., Goos, P., Braysy, O.: Efficient GRASP+VND and GRASP+VNS metaheuristics for the traveling repairman problem. J. Operations Research 9(2), 189–209 (2011)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Silva, M., Subramanian, A., Vidal, T., Ochi, L.: A simple and effective metaheuristic for the Minimum Latency Problem. J. Operations Research 221(3), 513–520 (2012)CrossRefzbMATHGoogle Scholar
  13. 13.
    Rosenkrantz, D.J., Stearns, R.E., Lewis, P.M.: An analysis of several heuristics for the traveling salesman problem. SIAM J. Comput. (6), 563–581 (1977)Google Scholar
  14. 14.
    Polyak, B.T.: Minimization of unsmooth functionals. U.S.S.R. Computational Mathematics and Mathematical Physis 9(3), 14–29 (1969)CrossRefGoogle Scholar
  15. 15.
    Wu, B.Y., Huang, Z.-N., Zhan, F.-J.: Exact algorithms for the minimum latency problem. Inform. Proc. Letters 92(6), 303–309 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
  17. 17.

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Hanoi University of Science and TechnologyHa NoiVietnam

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