Inverse minimum flow problem under the weighted sum-type Hamming distance

In this paper, we consider the inverse minimum flow problem under the weighted sum-type Hamming distance, where the lower and upper bounds for the arcs should be changed as little as possible under the weighted sum-type Hamming distance such that a given feasible flow becomes a minimum flow. Two models are considered: the unbounded case and the general bounded case. We present their respective combinatorial algorithms that both run in O(nm) time in terms of the minimum cut method. Computational examples are presented to illustrate our algorithms.