Capacity inverse minimum cost flow problem

Given a directed graph G = (N, A) with arc capacities uij and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector ˆu for the arc set A such that a given feasible solution ˆx is optimal with respect to the modified capacities. Among all capacity vectors ˆu satisfying this condition, we would like to find one with minimum ˆu − u value. We consider two distance measures for ˆu − u , rectilinear (L1) and chebyshev (L∞) distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is 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 also propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs.

• Type: Journal
• Year: 2010
• Where: JCO
• Authors: Çigdem Güler, Horst W. Hamacher
• Keywords: Capacity Vectors, Cost flow Problem, JCO 2010, flow Problem