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Inverse minimum cost flow problems under the weighted Hamming distance

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  • Jiang, Yiwei
  • Liu, Longcheng
  • Wu, Biao
  • Yao, Enyu

Abstract

Given a network N(V, A, u, c) and a feasible flow x0, an inverse minimum cost flow problem is to modify the cost vector as little as possible to make x0 form a minimum cost flow of the network. The modification can be measured by different norms. In this paper, we consider the inverse minimum cost flow problems, where the modification of the arcs is measured by the weighted Hamming distance. Both the sum-type and the bottleneck-type cases are considered. For the former, it is shown to be APX-hard due to the weighted feedback arc set problem. For the latter, we present a strongly polynomial algorithm which can be done in O(n · m2).

Suggested Citation

  • Jiang, Yiwei & Liu, Longcheng & Wu, Biao & Yao, Enyu, 2010. "Inverse minimum cost flow problems under the weighted Hamming distance," European Journal of Operational Research, Elsevier, vol. 207(1), pages 50-54, November.
    • Handle: RePEc:eee:ejores:v:207:y:2010:i:1:p:50-54
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    References listed on IDEAS

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    1. Binwu Zhang & Jianzhong Zhang & Yong He, 2005. "The Center Location Improvement Problem Under the Hamming Distance," Journal of Combinatorial Optimization, Springer, vol. 9(2), pages 187-198, March.
    2. Longcheng Liu & Enyu Yao, 2007. "A Weighted Inverse Minimum Cut Problem Under The Bottleneck Type Hamming Distance," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 24(05), pages 725-736.
    3. Yong He & Binwu Zhang & Enyu Yao, 2005. "Weighted Inverse Minimum Spanning Tree Problems Under Hamming Distance," Journal of Combinatorial Optimization, Springer, vol. 9(1), pages 91-100, February.
    4. 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.
    5. 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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    2. Souza, J.W.G. & Pereira, H.B.B. & Santos, A.A.B. & Senna, V. & Moret, M.A., 2014. "A new proposal for analyzing combustion process stability based on the Hamming distance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 413(C), pages 301-306.
    3. Binwu Zhang & Xiucui Guan & Panos M. Pardalos & Chunyuan He, 2018. "An Algorithm for Solving the Shortest Path Improvement Problem on Rooted Trees Under Unit Hamming Distance," Journal of Optimization Theory and Applications, Springer, vol. 178(2), pages 538-559, August.

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