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
for the arc set A such that a given feasible flow
is optimal with respect to the modified capacities. Among all capacity vectors
satisfying this condition, we would like to find one with minimum
We consider two distance measures for
, 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
-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.
Inverse problems Network flows Minimum cost flows
This is a preview of subscription content, log in to check access.
Ausiello G, Crescenzi P, Gambosi G, Kann V, Marchetti-Spaccamela A, Protasi M (1999) Complexity and approximation: combinatorial optimization problems and their approximability properties. Springer, Berlin
Kann V (1992) On the approximability of NP-complete optimization problems. PhD thesis, Department of Numerical Analysis and Computing Science, Royal Institute of Technology, Sweden
Karp RM (1972) Reducibility among combinatorial problems. In: Miller RE, Thatcher JW (eds) Complexity of computer computations, pp 85–103
Klingman D, Napier A, Stutz J (1974) NETGEN: a program for generating large scale capacitated assignment, transportation, and minimum cost flow network problems. Manage Sci 20:814–820