# American Institute of Mathematical Sciences

January  2012, 8(1): 41-49. doi: 10.3934/jimo.2012.8.41

## A note on the subtree ordered median problem in networks based on nestedness property

 1 Faculty of Management and Administration, Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau SAR, China 2 Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China 3 Department of Logistics and Maritime Studies, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong SAR, China

Received  January 2011 Revised  May 2011 Published  November 2011

The nestedness property has become an increasingly important means for devising efficient algorithms for network location problems. In this paper we prove that the nestedness property holds for the tactical continuous, and strategic discrete and continuous subtree location problems in a tree network with the ordered median objective, where the $\lambda$-weights take at most two different values. These results extend some existing results in the literature. With these nestedness results, we solve the problems in polynomial time. Finally we pose an open problem on identifying the nestedness property for the $(k_1,k_2)$-trimmed problem.
Citation: Huajun Tang, T. C. Edwin Cheng, Chi To Ng. A note on the subtree ordered median problem in networks based on nestedness property. Journal of Industrial & Management Optimization, 2012, 8 (1) : 41-49. doi: 10.3934/jimo.2012.8.41
##### References:
 [1] J. N. Hooker, R. S. Garfinkel and C. K. Chen, Finite dominating sets for network location problems,, Operations Research, 39 (1991), 100.  doi: 10.1287/opre.39.1.100.  Google Scholar [2] J. Kalcsics, S. Nickel and J. Puerto, Multi-facility ordered median problems: A further analysis,, Networks, 41 (2003), 1.  doi: 10.1002/net.10053.  Google Scholar [3] E. Minieka, The optimal location of a path or tree in a tree network,, Networks, 15 (1985), 309.  doi: 10.1002/net.3230150304.  Google Scholar [4] S. Nickel, and J. Puerto, "Location Theory: A Unified Approach,", 1st edition, (2005).   Google Scholar [5] W. Ogryczak and A. Tamir, Minimizing the sum of $k$ largest functions in linear time,, Information Processing Letters, 85 (2003), 117.  doi: 10.1016/S0020-0190(02)00370-8.  Google Scholar [6] J. Puerto, and A. Tamir, Locating tree-shaped facilities using the ordered median objective,, Mathematical Programming, 102 (2005), 313.  doi: 10.1007/s10107-004-0547-2.  Google Scholar [7] A. Tamir, J. Puerto and D. Pérez-Brito, The centdian subtree on tree networks,, Discrete Applied Mathematics, 18 (2002), 263.  doi: 10.1016/S0166-218X(01)00199-8.  Google Scholar [8] H. J. Tang, T. C. E. Cheng and C. T. Ng, Finite dominating sets for the multi-facility ordered median problem in networks and algorithmic applications,, Computers & Industrial Engineering, 57 (2009), 707.  doi: 10.1016/j.cie.2009.01.015.  Google Scholar [9] H. J. Tang, T. C. E. Cheng and C. T. Ng, Multi-facility ordered median problems in directed networks,, Journal of Systems Science and Complexity, 24 (2011), 61.  doi: 10.1007/s11424-011-9327-2.  Google Scholar [10] P. M. Vaidya, An algorithm for linear programming which requires $O((m+n)n^2+(m+n)^{1.5}nL)$ arithmetic operations,, Mathematical Programming, 47 (1990), 175.  doi: 10.1007/BF01580859.  Google Scholar [11] B.-F. Wang, Efficient parallel algorithms for optimally locating a path and a tree of a specified length in a weighted tree network,, Journal of Algorithms, 34 (2000), 90.  doi: 10.1006/jagm.1999.1020.  Google Scholar

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##### References:
 [1] J. N. Hooker, R. S. Garfinkel and C. K. Chen, Finite dominating sets for network location problems,, Operations Research, 39 (1991), 100.  doi: 10.1287/opre.39.1.100.  Google Scholar [2] J. Kalcsics, S. Nickel and J. Puerto, Multi-facility ordered median problems: A further analysis,, Networks, 41 (2003), 1.  doi: 10.1002/net.10053.  Google Scholar [3] E. Minieka, The optimal location of a path or tree in a tree network,, Networks, 15 (1985), 309.  doi: 10.1002/net.3230150304.  Google Scholar [4] S. Nickel, and J. Puerto, "Location Theory: A Unified Approach,", 1st edition, (2005).   Google Scholar [5] W. Ogryczak and A. Tamir, Minimizing the sum of $k$ largest functions in linear time,, Information Processing Letters, 85 (2003), 117.  doi: 10.1016/S0020-0190(02)00370-8.  Google Scholar [6] J. Puerto, and A. Tamir, Locating tree-shaped facilities using the ordered median objective,, Mathematical Programming, 102 (2005), 313.  doi: 10.1007/s10107-004-0547-2.  Google Scholar [7] A. Tamir, J. Puerto and D. Pérez-Brito, The centdian subtree on tree networks,, Discrete Applied Mathematics, 18 (2002), 263.  doi: 10.1016/S0166-218X(01)00199-8.  Google Scholar [8] H. J. Tang, T. C. E. Cheng and C. T. Ng, Finite dominating sets for the multi-facility ordered median problem in networks and algorithmic applications,, Computers & Industrial Engineering, 57 (2009), 707.  doi: 10.1016/j.cie.2009.01.015.  Google Scholar [9] H. J. Tang, T. C. E. Cheng and C. T. Ng, Multi-facility ordered median problems in directed networks,, Journal of Systems Science and Complexity, 24 (2011), 61.  doi: 10.1007/s11424-011-9327-2.  Google Scholar [10] P. M. Vaidya, An algorithm for linear programming which requires $O((m+n)n^2+(m+n)^{1.5}nL)$ arithmetic operations,, Mathematical Programming, 47 (1990), 175.  doi: 10.1007/BF01580859.  Google Scholar [11] B.-F. Wang, Efficient parallel algorithms for optimally locating a path and a tree of a specified length in a weighted tree network,, Journal of Algorithms, 34 (2000), 90.  doi: 10.1006/jagm.1999.1020.  Google Scholar
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