Geodesic
The Inverse Maximum Flow Problem With Lower and Upper Bounds for the Flow
Yugoslav journal of operations research, Tome 18 (2008) no. 1, p. 13 .

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The general inverse maximum flow problem (denoted GIMF) is considered, where lower and upper bounds for the flow are changed so that a given feasible flow becomes a maximum flow and the distance (considering $l_1$ norm) between the initial vector of bounds and the modified vector is minimum. Strongly and weakly polynomial algorithms for solving this problem are proposed. In the paper it is also proved that the inverse maximum flow problem where only the upper bound for the flow is changed (IMF) is a particular case of the GIMF problem.
Classification : 90B10 05C85 90C35
Keywords: Inverse problems, maximum flow, minimum cut, residual network, graph search 
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Adrian Deaconu. The Inverse Maximum Flow Problem With Lower and Upper Bounds for the Flow. Yugoslav journal of operations research, Tome 18 (2008) no. 1, p. 13 . http://geodesic.mathdoc.fr/item/YJOR_2008_18_1_a1/