Abstract
A dynamic network introduced by Ford and Fulkerson is a directed graph with capacities and transit times on its arcs. The quickest transshipment problem is one of the most fundamental problems in dynamic networks. In this problem, we are given sources and sinks. Then the goal of this problem is to find a minimum time limit such that we can send the right amount of flow from sources to sinks. In this paper, we introduce a variant of this problem called the mixed evacuation problem. This problem models an emergent situation in which people can evacuate on foot or by car. The goal is to organize such a mixed evacuation so that an efficient evacuation can be achieved. In this paper, we study this problem from the theoretical and practical viewpoints. In the first part, we prove the polynomial-time solvability of this problem in the case where the number of sources and sinks is not large, and also prove the polynomial-time solvability and computational hardness of its variants with integer constraints. In the second part, we apply our model to the case study of Minabe town in Wakayama prefecture, Japan.
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Notes
This result was cited by Hoppe and Tardos (2000) as personal communication.
References
Baumann N, Skutella M (2009) Earliest arrival flows with multiple sources. Math Oper Res 34(2):499–512
Cook W J, Cunningham W H, Pulleyblank W R, Schrijver A (1998) Combinatorial optimization. Wiley, Hoboken
Ford LR Jr, Fulkerson DR (1958) Constructing maximal dynamic flows from static flows. Oper Res 6(3):419–433
Ford LR Jr, Fulkerson R (1962) Flows in networks. Princeton University Press, Princeton
Fujishige S (2005) Submodular functions and optimization volume 58 of annals of discrete mathematics. Elsevier, Amsterdam
Garey MR, Johnson DS (1979) Computers and Intractability: a guide to the theory of NP-completeness. W. H Freeman and Company, London
Grötschel M, Lovász L, Schrijver A (1993) Geometric algorithms and combinatorial optimization, 2nd edn. Springer, Berlin
Hoppe B, Tardos É (2000) The quickest transshipment problem. Math Oper Res 25(1):36–62
Iwata S, Fleischer L, Fujishige S (2001) A combinatorial strongly polynomial algorithm for minimizing submodular functions. J ACM 48(4):761–777
Khachiyan LG (1980) Polynomial algorithms in linear programming. USSR Comput Math Math Phys 20(1):53–72
Li C-L, McCormick ST, Simchi-Levi D (1992) Finding disjoint paths with different path-costs: complexity and algorithms. Networks 22(7):653–667
Lin M, Jaillet P (2015) On the quickest flow problem in dynamic networks—a parametric min-cost flow approach. In: Proceedings of the 26th annual ACM-SIAM symposium on discrete algorithms, pp 1343–1356
Orlin JB (1993) A faster strongly polynomial minimum cost flow algorithm. Oper Res 41(2):338–350
Saho M, Shigeno M (2017) Cancel-and-tighten algorithm for quickest flow problems. Networks 69(2):179–188
Schlöter M, Skutella M (2017) Fast and memory-efficient algorithms for evacuation problems. In: Proceedings of the 28th annual ACM-SIAM symposium on discrete algorithms, pp 821–840
Schrijver A (2000) A combinatorial algorithm minimizing submodular functions in strongly polynomial time. J Comb Theory Ser B 80(2):346–355
Schrijver A (2003) Combinatorial optimization: polyhedra and efficiency. Springer, Berlin
(2016) The headquarters for earthquake research promotion. Evaluation of long-term probability of active fault and subduction-zone earthquake occurrence. http://www.jishin.go.jp/main/choukihyoka/ichiran.pdf (in Japanese)
Acknowledgements
This research was the result of the joint research with CSIS, the University of Tokyo (No. 573) and used the following data: Digital Road Map Database extended version 2013 provided by Sumitomo Electric Industries, Ltd and Zmap TOWN II 2008/09 Shapefile Wakayama prefecture provided by Zenrin Co. Ltd. Yuya Higashikawa, Naoki Katoh, and Atsushi Takizawa were supported by JSPS Grant-in-Aid for Scientific Research(A) (25240004) and by JST CREST Grant Number JPMJCR1402, Japan. Naoyuki Kamiyama was supported by JST PRESTO Grant Numbers JPMJPR14E1, Japan.
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An earlier version of this paper has appeared in Proceedings of the 10th Annual International Conference on Combinatorial Optimization and Applications (COCOA), volume 10043 of Lecture Notes in Computer Science, pages 18–32, 2016.
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Hanawa, Y., Higashikawa, Y., Kamiyama, N. et al. The mixed evacuation problem. J Comb Optim 36, 1299–1314 (2018). https://doi.org/10.1007/s10878-017-0237-7
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DOI: https://doi.org/10.1007/s10878-017-0237-7