Abstract
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.
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The research has been partially supported by the Deutsche Forschungsgemeinschaft (DFG), Grant HA 1737/7 “Algorithmik großer und komplexer Netzwerke” and by the Rheinland-Pfalz cluster of excellence “Dependable adaptive systems and mathematical modeling”.
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Güler, Ç., Hamacher, H.W. Capacity inverse minimum cost flow problem. J Comb Optim 19, 43–59 (2010). https://doi.org/10.1007/s10878-008-9159-8
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DOI: https://doi.org/10.1007/s10878-008-9159-8