Abstract
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.
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References
Ben-Moshe, B., Bhattachary, B., Shi, Q., Tamir, A.: Efficient algorithms for center problems in cactus networks. Theor. Comput. Sci. 378, 237–252 (2007)
Ben-Moshe, B., Dvir, A., Segal, M., Tamir, A.: Centdian computation in cactus graphs. J. Graph Algorithms Appl. 16, 199–224 (2012)
Burkard, R.E., Krarup, J.: A linear algorithm for the pos/neg-weighted 1-median problem on a cactus. Computing 60, 193–215 (1998)
Burkard, R.E., Fathali, J., Kakhki, H.T.: The \(p\)-maxian problem on a tree. Oper. Res. Lett. 35, 331–335 (2007)
Cole, R.: Slowing down sorting networks to obtain faster algorithms. J. Assoc. Comput. Math. 34, 168–177 (1987)
Gavish, B., Sridhar, S.: Computing the 2-median on tree networks in \(O(n \log n)\) time. Networks 26, 305–317 (1995)
Goldman, A.J.: Optimal center location in simple networks. Transp. Sci. 5, 539–560 (1971)
Handler, G.Y.: Minimax location of a facility in an undirected tree graph. Transp. Sci. 7, 287–293 (1973)
Hua, L.K.: Application off mathematical models to wheat harvesting. Chin. Math. 2, 539–560 (1962)
Kalcsics, J., Nickel, S., Puerto, J., Tamir, A.: Algorithmic approach for the ordered median problems. Oper. Res. Lett. 30, 149–158 (2002)
Kang, L., Bai, C., Shang, E., Nguyen, K.: The 2-maxian problem on cactus graphs. Discrete Optim. 13, 16–22 (2014)
Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems, I. The p-centers. SIAM J. Appl. Math. 37, 513–538 (1979)
Kariv, O., Hakimi, S.L.: An algorithmic approach to network location problems, II. The p-medians. SIAM J. Appl. Math. 37, 536–560 (1979)
Megiddo, N.: Linear-time algorithms for linear programming in \({\mathbb{R}}^3\) and related problems. SIAM J. Comput. 12, 759–776 (1983)
Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. J. Assoc. Comput. Math. 30, 852–865 (1983)
Nickel, S., Puerto, J.: Location Theory—A Unified Approach. Springer, Berlin (2004)
Tamir, A.: An \(O(pn^2)\) algorithm for the \(p\)-median and related problems on tree graphs. Oper. Res. Lett. 19, 59–64 (1994)
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The authors would like to acknowledge the anonymous referees for valuable comments which helped to improve the paper significantly.
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Pham, V.H., Tam, N.C. A combinatorial algorithm for the ordered 1-median problem on cactus graphs. OPSEARCH 56, 780–789 (2019). https://doi.org/10.1007/s12597-019-00402-2
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DOI: https://doi.org/10.1007/s12597-019-00402-2