Capacity inverse minimum cost flow problems under the weighted Hamming distance

Given an instance of the minimum cost flow problem, a version of the corresponding inverse problem, called the capacity inverse problem, is to modify the upper and lower bounds on arc flows as little as possible so that a given feasible flow x becomes optimal to the modified minimum cost flow problem. The modifications can be measured by different distances. Here, we consider the capacity inverse problem under the bottleneck-type and the sum-type weighted Hamming distances. In the bottleneck-type case, the binary search technique is applied to present an algorithm for solving the problem in O(nm log n) time. In the sum-type case, it is shown that the inverse problem is strongly NP-hard even on bipartite networks.