An inverse optimization problem is defined as follows: Let
denote the set of feasible solutions of an optimization problem
be a specified cost (capacity) vector , and
. We want to perturb the cost (capacity) vector
becomes an optimal solution of
with respect to the cost (capacity) vector
, and to minimize some objective functions. In this paper, we consider the weighted inverse minimum cut problem under the sum-type Hamming distance. First, we show the general case is NP-hard. Second we present a combinatorial algorithm that run in strongly polynomial time to solve a special case.