Abstract
In this paper we consider strongly polynomial variations of the auction algorithm for the single origin/many destinations shortest path problem. These variations are based on the idea of graph reduction, that is, deleting unnecessary arcs of the graph by using certain bounds naturally obtained in the course of the algorithm. We study the structure of the reduced graph and we exploit this structure to obtain algorithms withO (n min{m, n logn}) andO(n 2) running time. Our computational experiments show that these algorithms outperform their closest competitors on randomly generated dense all destinations problems, and on a broad variety of few destination problems.
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References
D.P. Bertsekas, “A Distributed Algorithm for the Assignment Problems,” Lab. for Information and Decision Systems Working Paper, M.I.T., Cambridge, MA, March 1979.
D.P. Bertsekas, “A New Algorithm for the Assignment Problem,” Math. Programming, vol. 21, pp. 152–171, 1981.
D.P. Bertsekas, “The Auction Algorithm for Assignment and Other Network Flow Problems: A Tutorial,” Interfaces, vol. 20, pp. 133–149, 1990.
D.P. Bertsekas, “The Auction Algorithm for Shortest Paths,” SIAM J. on Optimization, vol. 1, pp. 425–447, 1991.
D.P. Bertsekas, “Linear Network Optimization: Algorithms and Codes,” M.I.T. Press, Cambridge, MA, 1991.
Z. Galil and A. Naamad, “O (VE log2 V) Algorithms for the Maximum Flow Problem,” J. of Comput. Sys. Sci., vol. 21, pp. 203–217, 1980.
G. Gallo and S. Pallottino, “Shortest Path Methods: A Unified Approach,” Math. Programming Study, vol. 26, pp. 38–64, 1986.
G. Gallo and S. Pallottino, “Shortest Path Algorithms,” Annals of Operations Research, vol. 7, pp. 3–79 1988.
D. Klingman, A. Napier, and J. Stutz, “NETGEN—A Program for Generating Large Scale (Un) Capacitated Assignment, Transportation, and Minimum Cost Flow Network Problems,” Management Science, vol. 20, pp. 814–822, 1974.
C.H. Papadimitriou and K. Steiglitz, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Englewood Cliffs, N.J., 1982.
S. Pallottino and M.G. Scutellà, “Strongly Polynomial Algorithms for Shortest Paths,” Ricerca Operativa, vol 60, pp. 33–53, 1991.
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Research supported by NSF under Grant No. DDM-8903385, by the ARO under Grant DAAL03-86-K-0171, by a CNR-GNIM grant, and by a Fullbright grant
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Bertsekas, D.P., Pallottino, S. & Scutellà, M.G. Polynomial auction algorithms for shortest paths. Comput Optim Applic 4, 99–125 (1995). https://doi.org/10.1007/BF01302891
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DOI: https://doi.org/10.1007/BF01302891