Minimax Regret 1-Median Problem in Dynamic Path Networks

  • Yuya Higashikawa
  • Siu-Wing Cheng
  • Tsunehiko Kameda
  • Naoki Katoh
  • Shun Saburi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9843)

Abstract

This paper considers the minimax regret 1-median problem in dynamic path networks. In our model, we are given a dynamic path network consisting of an undirected path with positive edge lengths, uniform positive edge capacity, and nonnegative vertex supplies. Here, each vertex supply is unknown but only an interval of supply is known. A particular assignment of supply to each vertex is called a scenario. Given a scenario s and a sink location x in a dynamic path network, let us consider the evacuation time to x of a unit supply given on a vertex by s. The cost of x under s is defined as the sum of evacuation times to x for all supplies given by s, and the median under s is defined as a sink location which minimizes this cost. The regret for x under s is defined as the cost of x under s minus the cost of the median under s. Then, the problem is to find a sink location such that the maximum regret for all possible scenarios is minimized. We propose an \(O(n^3)\) time algorithm for the minimax regret 1-median problem in dynamic path networks with uniform capacity, where n is the number of vertices in the network.

Keywords

Minimax regret Sink location Dynamic flow Evacuation planning 

References

  1. 1.
    Arumugam, G.P., Augustine, J., Golin, M.J., Srikanthan, P.: A polynomial time algorithm for minimax-regret evacuation on a dynamic path. CoRR abs/1404.5448. arXiv:1404.5448
  2. 2.
    Averbakh, I., Berman, O.: Algorithms for the robust \(1\)-center problem on a tree. Eur. J. Oper. Res. 123(2), 292–302 (2000)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bhattacharya, B., Kameda, T.: A linear time algorithm for computing minmax regret 1-median on a tree. In: Gudmundsson, J., Mestre, J., Viglas, T. (eds.) COCOON 2012. LNCS, vol. 7434, pp. 1–12. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  4. 4.
    Bhattacharya, B., Kameda, T.: Improved algorithms for computing minmax regret 1-sink and 2-sink on path network. In: Zhang, Z., Wu, L., Xu, W., Du, D.-Z. (eds.) COCOA 2014. LNCS, vol. 8881, pp. 146–160. Springer, Heidelberg (2014)Google Scholar
  5. 5.
    Bhattacharya, B., Kameda, T., Song, Z.: A linear time algorithm for computing minmax regret \(1\)-median on a tree network. Algorithmica 62, 1–20 (2013)MathSciNetMATHGoogle Scholar
  6. 6.
    Brodal, G.S., Georgiadis, L., Katriel, I.: An \(O(n \log n)\) version of the Averbakh-Berman algorithm for the robust median of a tree. Oper. Res. Lett. 36(1), 14–18 (2008)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Chen, B., Lin, C.: Minmax-regret robust 1-median location on a tree. Networks 31(2), 93–103 (1998)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cheng, S.-W., Higashikawa, Y., Katoh, N., Ni, G., Su, B., Xu, Y.: Minimax regret 1-sink location problems in dynamic path networks. In: Chan, T.-H.H., Lau, L.C., Trevisan, L. (eds.) TAMC 2013. LNCS, vol. 7876, pp. 121–132. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  9. 9.
    Conde, E.: A note on the minmax regret centdian location on trees. Oper. Res. Lett. 36(2), 271–275 (2008)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dyer, M.E.: Linear time algorithms for two- and three-variable linear programs. SIAM J. Comput. 13(1), 31–45 (1984)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Ford Jr., L.R., Fulkerson, D.R.: Constructing maximal dynamic flows from static flows. Oper. Res. 6, 419–433 (1958)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Higashikawa, Y.: Studies on the Space Exploration and the Sink Location under Incomplete Information towards Applications to Evacuation Planning. Doctoral Dissertation, Kyoto University (2014)Google Scholar
  13. 13.
    Higashikawa, Y., Augustine, J., Cheng, S.W., Katoh, N., Ni, G., Su, B., Xu, Y.: Minimax regret 1-sink location problem in dynamic path networks. Theor. Comput. Sci. 588, 24–36 (2015). doi:10.1016/j.tcs.2014.02.010 MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Higashikawa, Y., Golin, M.J., Katoh, N.: Minimax regret sink location problem in dynamic tree networks with uniform capacity. In: Pal, S.P., Sadakane, K. (eds.) WALCOM 2014. LNCS, vol. 8344, pp. 125–137. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  15. 15.
    Higashikawa, Y., Golin, M.J., Katoh, N.: Multiple sink location problems in dynamic path networks. In: Gu, Q., Hell, P., Yang, B. (eds.) AAIM 2014. LNCS, vol. 8546, pp. 149–161. Springer, Heidelberg (2014)Google Scholar
  16. 16.
    Higashikawa, Y., Golin, M.J., Katoh, N.: Minimax regret sink location problem in dynamic tree networks with uniform capacity. J. Graph Algorithms Appl. 18(4), 539–555 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Higashikawa, Y., Golin, M.J., Katoh, N.: Multiple sink location problems in dynamic path networks. Theor. Comput. Sci. 607(Part 1), 2–15 (2015). doi:10.1016/j.tcs.2015.05.053 MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Mamada, S., Uno, T., Makino, K., Fujishige, S.: An \(O(n \log ^2 n)\) algorithm for the optimal sink location problem in dynamic tree networks. Discrete Appl. Math. 154(16), 2387–2401 (2006)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Wang, H.: Minmax regret 1-facility location on uncertain path networks. In: Cai, L., Cheng, S.-W., Lam, T.-W. (eds.) Algorithms and Computation. LNCS, vol. 8283, pp. 733–743. Springer, Heidelberg (2013)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Yuya Higashikawa
    • 1
    • 6
  • Siu-Wing Cheng
    • 2
  • Tsunehiko Kameda
    • 3
  • Naoki Katoh
    • 4
    • 6
  • Shun Saburi
    • 5
  1. 1.Department of Information and System EngineeringChuo UniversityTokyoJapan
  2. 2.Department of Computer Science and EngineeringThe Hong Kong University of Science and TechnologyHong KongChina
  3. 3.School of Computing ScienceSimon Fraser UniversityBurnabyCanada
  4. 4.Department of InformaticsKwansei Gakuin UniversitySandaJapan
  5. 5.Department of Architecture and Architectural EngineeringKyoto UniversityKyotoJapan
  6. 6.CREST, Japan Science and Technology Agency (JST)KawaguchiJapan

Personalised recommendations