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Locating tree-shaped facilities using the ordered median objective

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Abstract.

In this paper we consider the location of a tree-shaped facility S on a tree network, using the ordered median function of the weighted distances to represent the total transportation cost objective. This function unifies and generalizes the most common criteria used in location modeling, e.g., median and center. If there are n demand points at the nodes of the tree T=(V,E), this function is characterized by a sequence of reals, Λ=(λ 1, . . . ,λ n ), satisfying λ 1≥λ 2≥...≥λ n ≥0. For a given subtree S let X(S)={x 1, . . . ,x n } be the set of weighted distances of the n demand points from S. The value of the ordered median objective at S is obtained as follows: Sort the n elements in X(S) in nonincreasing order, and then compute the scalar product of the sorted list with the sequence Λ. Two models are discussed. In the tactical model, there is an explicit bound L on the length of the subtree, and the goal is to select a subtree of size L, which minimizes the above transportation cost function. In the strategic model the length of the subtree is variable, and the objective is to minimize the sum of the transportation cost and the length of the subtree. We consider both discrete and continuous versions of the tactical and the strategic models. We note that the discrete tactical problem is NP-hard, and we solve the continuous tactical problem in polynomial time using a Linear Programming (LP) approach. We also prove submodularity properties for the strategic problem. These properties allow us to solve the discrete strategic version in strongly polynomial time. Moreover the continuous version is also solved via LP. For the special case of the k-centrum objective we obtain improved algorithmic results using a Dynamic Programming (DP) algorithm and discretization and nestedness results.

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References

  1. Alstrup, S., Lauridsen, P.W., Sommerlund, P., Thorup, M.: Finding cores of limited length. In: Algorithms and Data Structures, Lecture Notes in Computer Science n. 1271, Dehne, F., Rau-Chaplin, A. Sack, J-R., Tamassia R. (eds.), Springer, 1997, pp. 45–54

  2. Becker, R.I., Perl, Y.: Finding the two-core of a tree. Discrete Appl. Math. 11, 103–113 (1985)

    MATH  MathSciNet  Google Scholar 

  3. Becker, R.I., Lari, I., Scozzari, A.: Efficient algorithms for finding the (k,l)-core of tree networks. Networks 40, 208–215 (2003)

    Article  MATH  Google Scholar 

  4. Boffey, B., Mesa, J.A.: A review of extensive facility location in networks. European J. Operational Res. 95, 592–600 (1996)

    MATH  Google Scholar 

  5. Bramel, J., Simchi-Levi, D.: The Logic of Logistics: Theory, Algorithms and Applications for Logistics Management. Springer 1997, Berlin

  6. Cho, G., Shaw, D.X.: A depth-first dynamic programming algorithm for the tree knapsack problem. INFORMS J. Comput. 9, 431–438 (1997)

    MathSciNet  MATH  Google Scholar 

  7. Faigle, U., Kern, W.: Computational complexity of some maximum average weight problems with precedence constraints. Operations Res. 42, 688–693 (1994)

    MATH  MathSciNet  Google Scholar 

  8. Fischetti, M., Hamacher, H.W., Jornsten, K., Maffioli, F.: Weighted k-cardinality trees: complexity and polyhedral structure. Networks 24, 11–21 (1994)

    MATH  MathSciNet  Google Scholar 

  9. Francis, R.L., Lowe, T.J., Tamir, A.: Aggregation error bounds for a class of location models. Operations Res. 48, 294–307 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  10. Francis, R.L., Lowe, T.J., Tamir, A.: Worst-case incremental analysis for a class of p-facility location problems. Networks 39, 139–143 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Groetschel, M., Lovasz, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer-Verlag 1993, Berlin

  12. Hakimi, S.L., Schmeichel, E.F., Labbe, M.: On locating path-or tree shaped facilities on networks. Networks 23, 543–555 (1993)

    MATH  MathSciNet  Google Scholar 

  13. Hedetniemi, S.M., Cockaine, E.J., Hedetniemi, S.T.: Linear algorithms for finding the Jordan center and path center of a tree. Transportation Sci. 15, 98–114 (1981)

    Article  MathSciNet  Google Scholar 

  14. Johnson, D.S., Niemi, K.A.: On knapsack, partitions and a new dynamic technique for trees. Math. Operations Res. 8, 1–14 (1983)

    MATH  Google Scholar 

  15. Kalcsics, J., Nickel, S., Puerto, J.: Multi-facility ordered median problems: A further analysis. Networks 41, 1–12 (2003)

    MATH  MathSciNet  Google Scholar 

  16. Kalcsics, J., Nickel, S., Puerto, J., Tamir, A.: Algorithmic results for ordered median problems defined on networks and the plane. Operations Res. Lett. 30, 149–158 (2002)

    MATH  MathSciNet  Google Scholar 

  17. Kim, T.U., Lowe, T.J., Tamir, A., Ward, J.E.: On the location of a tree-shaped facility. Networks 28, 167–175 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  18. McCormick, S.T.: Submodular function minimization. To appear as a chapter in Handbook on Discrete Optimization, edited by K. Aardal, G. Nemhauser and R. Weismantel

  19. Minieka, E.: Conditional centers and medians on a graph. Networks 10, 265–272 (1980)

    MathSciNet  Google Scholar 

  20. Minieka, E.: The optimal location of a path or tree in a tree network. Networks 15, 309–321 (1985)

    MATH  MathSciNet  Google Scholar 

  21. Minieka, E., Patel, N.H.: On finding the core of a tree with a specified length. J. Algorithms 4, 345–352 (1983)

    MATH  MathSciNet  Google Scholar 

  22. Morgan, C.A., Slater, J.P.: A linear algorithm for a core of a tree. J. Algorithms 1, 247–258 (1980)

    MATH  MathSciNet  Google Scholar 

  23. Nickel, S., Puerto, J.: A unified approach to network location problems. Networks 34, 283–290 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  24. Ogryczak, W., Tamir, A.: Minimizing the sum of the k largest functions in linear time. Inf. Processing Lett. 85, 117–122 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. Peng, S., Lo, W.: Efficient algorithms for finding a core of a tree with specified length. J. Algorithms 20, 445–458 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  26. Peng, S., Stephens, A.B., Yesha, Y.: Algorithms for a core and k-tree core of a tree. J. Algorithms 15, 143–159 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  27. Queyranne, N.M.: Minimizing symmetric submodular functions. Math. Programming 82, 3–12 (1998)

    Article  MathSciNet  Google Scholar 

  28. Rodríguez-Chía, A.M., Nickel, S., Puerto, J., Fernández, F.R.: A flexible approach to location problems. Math. Meth. Oper. Res. 51, 69–89 (2000)

    Article  MATH  Google Scholar 

  29. Shioura, A., Shigeno, M.: The tree center problems and the relationship with the bottleneck knapsack problems. Networks 29, 107–110 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  30. Slater, P.J.: Locating central paths in a graph. Transportation Sci 16, 1–18 (1982)

    MathSciNet  Google Scholar 

  31. Tamir, A.: A unifying location model on tree graphs based on submodularity properties. Discrete Appl. Math. 47, 275–283 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  32. Tamir, A.: An O(pn 2) algorithm for the p-median and related problems on tree graphs. Oper. Res. Lett. 19, 59–64 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  33. Tamir, A.: Fully polynomial approximation schemes for locating a tree-shaped facility: a generalization of the knapsack problem. Discrete Appl. Math. 87, 229–243 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  34. Tamir, A.: The k-centrum multi-facility location problem. Discrete Appl. Math. 109, 292–307 (2000)

    MathSciNet  Google Scholar 

  35. Tamir, A.: Sorting weighted distances with applications to objective function evaluations in single facility location problems. Oper. Res. Lett. 32, 249–257 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  36. Tamir, A., Lowe, T.J.: The generalized p-forest problem on a tree network. Networks 22, 217–230 (1992)

    MathSciNet  MATH  Google Scholar 

  37. Tamir, A., Puerto, J., Mesa, J.A., Rodriguez-Chia, A.M.: Conditional location of path and tree shaped facilities on trees. Technical Report, School of Mathematical Sciences, Tel-Aviv University, August 2001

  38. Tamir, A., Puerto, J., Perez-Brito, D.: The centdian subtree on tree networks. Discrete Appl. Math. 118, 263–278 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  39. Tardos, E.: A strongly polynomial algorithm to solve combinatorial linear programs. Oper. Res. 34, 250–256 (1986)

    MATH  MathSciNet  Google Scholar 

  40. Vaidya, P.M.: An algorithm for linear programming which requires O((m+n)n 2 + (m+n)1.5 nL) arithmetic operations. Math. Programming 47, 175–201 (1990)

  41. Wang, B-F: Efficient parallel algorithms for optimally locating a path and a tree of a specified length in a weighted tree network. J. Algorithms 34, 90–108 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  42. Wang, B-F: Finding a two-core of a tree in linear time. SIAM J. Discrete Math. 15, 193–210 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Facultad de Matemáticas, Universidad de Sevilla and Fundacion Centra, Spain

    J. Puerto

  2. School of Mathematical Sciences, Tel Aviv University, Israel

    A. Tamir

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  1. J. Puerto
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  2. A. Tamir
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Acknowledgement We would like to thank Noga Alon for the current version of the proof of Theorem 4.1, which simplifies our original proof significantly. J. Puerto also thanks the Spanish Ministerio de Ciencia y Tecnología through grant number BFM2001-2378 for partially supporting his research.

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Puerto, J., Tamir, A. Locating tree-shaped facilities using the ordered median objective. Math. Program. 102, 313–338 (2005). https://doi.org/10.1007/s10107-004-0547-2

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  • DOI: https://doi.org/10.1007/s10107-004-0547-2

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