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Capacity inverse minimum cost flow problem

Author

Listed:
  • Çiğdem Güler

    (University of Kaiserslautern)

  • Horst W. Hamacher

    (University of Kaiserslautern)

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.

Suggested Citation

  • Çiğdem Güler & Horst W. Hamacher, 2010. "Capacity inverse minimum cost flow problem," Journal of Combinatorial Optimization, Springer, vol. 19(1), pages 43-59, January.
  • Handle: RePEc:spr:jcomop:v:19:y:2010:i:1:d:10.1007_s10878-008-9159-8
    DOI: 10.1007/s10878-008-9159-8
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    References listed on IDEAS

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    1. D. Klingman & A. Napier & J. Stutz, 1974. "NETGEN: A Program for Generating Large Scale Capacitated Assignment, Transportation, and Minimum Cost Flow Network Problems," Management Science, INFORMS, vol. 20(5), pages 814-821, January.
    2. Jianzhong Zhang & Zhenhong Liu, 2002. "A General Model of Some Inverse Combinatorial Optimization Problems and Its Solution Method Under l ∞ Norm," Journal of Combinatorial Optimization, Springer, vol. 6(2), pages 207-227, June.
    3. Longcheng Liu & Jianzhong Zhang, 2006. "Inverse maximum flow problems under the weighted Hamming distance," Journal of Combinatorial Optimization, Springer, vol. 12(4), pages 395-408, December.
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    5. Clemens Heuberger, 2004. "Inverse Combinatorial Optimization: A Survey on Problems, Methods, and Results," Journal of Combinatorial Optimization, Springer, vol. 8(3), pages 329-361, September.
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    Cited by:

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    2. Susan Jia Xu & Mehdi Nourinejad & Xuebo Lai & Joseph Y. J. Chow, 2018. "Network Learning via Multiagent Inverse Transportation Problems," Service Science, INFORMS, vol. 52(6), pages 1347-1364, December.

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