Journal of Combinatorial Optimization

, Volume 19, Issue 1, pp 43–59 | Cite as

Capacity inverse minimum cost flow problem

  • Çiğdem Güler
  • Horst W. Hamacher


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.


Inverse problems Network flows Minimum cost flows 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany

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