Capacity inverse minimum cost flow problem

Given a directed graph G=(N,A) with arc capacities u ij and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector $\hat{u}$ for the arc set A such that a given feasible flow $\hat{x}$ is optimal with respect to the modified capacities. Among all capacity vectors $\hat{u}$ satisfying this condition, we would like to find one with minimum $\|\hat{u}-u\|$ value. We consider two distance measures for $\|\hat{u}-u\|$ , rectilinear (L 1) and Chebyshev (L ∞) distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is $\mathcal{NP}$ -hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.