pp 1–10 | Cite as

A combinatorial algorithm for the ordered 1-median problem on cactus graphs

  • Van Huy Pham
  • Nguyen Chi TamEmail author
Theoretical Article


Cactus graph is a graph in which any two simple cycles has at most one vertex in common. In this paper we address the ordered 1-median location problem on cactus graphs, a generalization of some popular location models such as 1-median, 1-center, and 1-centdian problems. For the case with non-decreasing multipliers, we show that there exists a cycle or an edge that contains an ordered 1-median. Based on this property, we develop a combinatorial algorithm that finds an ordered 1-median on a cactus in \(O(n^2\log n)\) time, where n is the number of vertices in the underlying cactus.


Location problem Ordered 1-median Cactus Convex 



The authors would like to acknowledge the anonymous referees for valuable comments which helped to improve the paper significantly.


  1. 1.
    Ben-Moshe, B., Bhattachary, B., Shi, Q., Tamir, A.: Efficient algorithms for center problems in cactus networks. Theor. Comput. Sci. 378, 237–252 (2007)CrossRefGoogle Scholar
  2. 2.
    Ben-Moshe, B., Dvir, A., Segal, M., Tamir, A.: Centdian computation in cactus graphs. J. Graph Algorithms Appl. 16, 199–224 (2012)CrossRefGoogle Scholar
  3. 3.
    Burkard, R.E., Krarup, J.: A linear algorithm for the pos/neg-weighted 1-median problem on a cactus. Computing 60, 193–215 (1998)CrossRefGoogle Scholar
  4. 4.
    Burkard, R.E., Fathali, J., Kakhki, H.T.: The \(p\)-maxian problem on a tree. Oper. Res. Lett. 35, 331–335 (2007)CrossRefGoogle Scholar
  5. 5.
    Cole, R.: Slowing down sorting networks to obtain faster algorithms. J. Assoc. Comput. Math. 34, 168–177 (1987)CrossRefGoogle Scholar
  6. 6.
    Gavish, B., Sridhar, S.: Computing the 2-median on tree networks in \(O(n \log n)\) time. Networks 26, 305–317 (1995)CrossRefGoogle Scholar
  7. 7.
    Goldman, A.J.: Optimal center location in simple networks. Transp. Sci. 5, 539–560 (1971)CrossRefGoogle Scholar
  8. 8.
    Handler, G.Y.: Minimax location of a facility in an undirected tree graph. Transp. Sci. 7, 287–293 (1973)CrossRefGoogle Scholar
  9. 9.
    Hua, L.K.: Application off mathematical models to wheat harvesting. Chin. Math. 2, 539–560 (1962)Google Scholar
  10. 10.
    Kalcsics, J., Nickel, S., Puerto, J., Tamir, A.: Algorithmic approach for the ordered median problems. Oper. Res. Lett. 30, 149–158 (2002)CrossRefGoogle Scholar
  11. 11.
    Kang, L., Bai, C., Shang, E., Nguyen, K.: The 2-maxian problem on cactus graphs. Discrete Optim. 13, 16–22 (2014)CrossRefGoogle Scholar
  12. 12.
    Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems, I. The p-centers. SIAM J. Appl. Math. 37, 513–538 (1979)CrossRefGoogle Scholar
  13. 13.
    Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems, II. The p-medians. SIAM J. Appl. Math. 37, 536–560 (1979)Google Scholar
  14. 14.
    Megiddo, N.: Linear-time algorithms for linear programming in \({\mathbb{R}}^3\) and related problems. SIAM J. Comput. 12, 759–776 (1983)CrossRefGoogle Scholar
  15. 15.
    Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. J. Assoc. Comput. Math. 30, 852–865 (1983)CrossRefGoogle Scholar
  16. 16.
    Nickel, S., Puerto, J.: Location Theory—A Unified Approach. Springer, Berlin (2004)Google Scholar
  17. 17.
    Tamir, A.: An \(O(pn^2)\) algorithm for the \(p\)-median and related problems on tree graphs. Oper. Res. Lett. 19, 59–64 (1994)CrossRefGoogle Scholar

Copyright information

© Operational Research Society of India 2019

Authors and Affiliations

  1. 1.AI Lab, Faculty of Information TechnologyTon Duc Thang UniversityHo Chi Minh CityVietnam
  2. 2.Department of Mathematics, Teacher CollegeCan Tho UniversityCan ThoVietnam

Personalised recommendations