Abstract
We consider some inverse min-max (or max-min) network problems. Such an inverse problem is to modify the weights with bound constraints so that a given feasible solution becomes an optimal solution of a min-max (or max-min) network problem, and the deviation of the weights, measured by the weighted l 1 norm or weighted l ∞ norm, is minimum. In this paper, we present strongly polynomial time algorithms to solve the inverse min-max spanning tree problem and the inverse maximum capacity path problem.
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References
Ahmed S, Guan YP (2005) The inverse optimal value problem. Math Program 102:91–110
Ahuja RK, Orlin JB (2001) Inverse optimization, part i: linear programming and general problem. Oper Res 35:771–783
Ahuja RK, Orlin JB (2000) A faster algorithm for the inverse spanning tree problem. J Algorithms 34:177–193
Burkard R, Pleschiutschnig C, Zhang JZ (2004) Inverse median problems. Discrete Optim 1:23–39
Burton D, Toint PhL (1992) On an instance of the inverse shortest paths problem. Math Program 53:45–61
Burton D, Toint PhL (1994) On the use of an inverse shortest paths algorithm for recovering linearly correlated costs. Math Program 63:1–22
Cai MC, Yang XG (1995) Inverse shortest path problems. Operations research and its applications. In: First international symposium ISORA’95 Beijing PR China
Cai MC, Yang XG, Zhang J (1999) The complexity analysis of the inverse center location problem. J Global Optim 15:213–218
Camerini PM (1978) The min-max spanning tree problem and some extensions. Inf Proc Lett 7:10–14
Hao J, Orlin JB (1994) A faster algorithm for finding the minimum cut in a directed graph. J Algorithms 17:424–446
Heuburger C (2004) Inverse optimization, a survey on problems, methods, and results. J Comb Optim 8:329–361
Hochbaum DS (2003) Efficient algorithms for the inverse spanning tree problem. Oper Res 51:785–797
He Y, Zhang B, Yao E (2005) Weighted inverse minimum spanning tree problems under Hamming distance. J Comb Optim 9:91–100
Iyengar G, Kang WM (2005) Inverse conic programming with applications. Oper Res Lett 33:319–330
Liu LC, He Y (2006) Inverse minimum spanning tree problem and reverse shortest-path problem with discrete values. Prog Nat Sci 16:649–655
Nagamochi H, Ibaraki T (1992) Computing edge-connectivity in multigraphs and capacitated graphs. SIAM J Discrete Math 5:54–66
Orlin JB (2003) Inverse optimization and partial inverse optimization. PPT presentation on Optimization Day Columbia University November 3
Schrijver A (2003) Combinatorial optimization: polyhedra and efficiency. Springer-Verlag Berlin and Heidelberg
Sokkalingam PT, Ahuja RK, Orlin JB (1999) Solving inverse spanning tree problems through network flow techniques. Oper Res 47:291–298
Xu S, Zhang J (1995) An inverse problem of the weighted shortest path problem. Japan J Ind Appl Math 12:47–59
Yang C, Zhang J (1998) Inverse maximum capacity problems. OR Spektrum 20:97–100
Yang C, Zhang J (1999) Two general methods for inverse optimization problems. Appl Math Lett 12:69–72
Zhang J, Liu Z (1999) A further study on inverse linear programming problems. J Comput Appl Math 106:345–359
Zhang J, Ma Z (1996) A network flow method for solving some inverse combinatorial optimization problems. Optimization 37:59–72
Zhang J, Xu S, Ma Z (1997) An algorithm for inverse minimum spanning tree problem. Optim Meth Soft 8:69–84
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Yang, X., Zhang, J. Some inverse min-max network problems under weighted l 1 and l ∞ norms with bound constraints on changes. J Comb Optim 13, 123–135 (2007). https://doi.org/10.1007/s10878-006-9016-6
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DOI: https://doi.org/10.1007/s10878-006-9016-6