Abstract
Given a network N(V, A, u, c) and a feasible flow \(x^0\), the inverse minimum cost flow problem is to modify the capacity vector or the cost vector as little as possible to make \(x^0\) form a minimum cost flow of the network. The modification can be measured by different norms. In this paper, we consider the capacity inverse minimum cost flow problems under the weighted Hamming distance, where we use the weighted Hamming distance to measure the modification of the arc capacities. 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 \(O(n\cdot m^2)\) time.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network flows: Theory, Algorithms and Applications. Pretice Hall, New York (1993)
Ahuja, R.K., Orlin, J.B.: Inverse optimization. Oper. Res. 49, 771–783 (2001)
Burton, D., Toint, P.L.: On an instance of the inverse shortest paths problem. Math. Program. 53, 45–61 (1992)
Duin, C.W., Volgenant, A.: Some inverse optimization problems under the Hamming distance. Eur. J. Oper. Res. 170, 887–899 (2006)
G\(\ddot{u}\)ler, C., Hamacher, H.W.: Capacity inverse minimum cost flow problem. J. Comb. Optim. 19, 43–59 (2010)
He, Y., Zhang, B.W., Yao, E.Y.: Weighted inverse minimum spanning tree problems under Hamming distance. J. Comb. Optim. 9, 91–100 (2005)
Heuberger, C.: Inverse optimization: a survey on problems, methods, and results. J. Comb. Optim. 8, 329–361 (2004)
Jiang, Y.W., Liu, L.C., Wu, B., Yao, E.Y.: Inverse minimum cost flow problems under the weighted Hamming distance. Eur. J. Oper. Res. 207, 50–54 (2010)
Kann, V.:On the approximability of NP-complete optimization problems. Ph.D thesis, Department of Numerical Analysis and Computing Science, Royal Institute of Technology, Sweden (1992)
Liu, L.C., Zhang, J.Z.: Inverse maximum flow problems under the weighted Hamming distance. J. Comb. Optim. 12, 395–408 (2006)
Tayyebi, J., Aman, M.: Note on “Inverse minimum cost flow problems under the weighted Hamming distance”. Eur. J. Oper. Res. 234, 916–920 (2014)
Yang, X.G., Zhang, J.Z.: Inverse sorting problem by minimizing the total weighted number of changes and partial inverse sorting problems. Comput. Optim. Appl. 36, 55–66 (2007)
Zhang, B.W., Zhang, J.Z., He, Y.: The center location improvement problem under the Hamming distance. J. Comput. Optim. 9, 187–198 (2005)
Zhang, B.W., Zhang, J.Z., He, Y.: Constrained inverse minimum spanning tree problems under the bottleneck-type Hamming distance. J. Global Optim. 34, 467–474 (2006)
Zhang, J.Z., Liu, Z.: Calculating some inverse linear programming problems. J. Comput. Appl. Math. 72, 261–273 (1996)
Acknowledgments
The authors wish to thank the anonymous referees whose valuable comments allowed us to improve the paper. This research is supported by the National Natural Science Foundation of China (Grant No. 11001232),Fujian Provincial Natural Science Foundation of China (Grant No. 2012J01021) and Fundamental Research Funds for the Central Universities (Grant No. 2010121004).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Liu, L., Yao, E. Capacity inverse minimum cost flow problems under the weighted Hamming distance. Optim Lett 10, 1257–1268 (2016). https://doi.org/10.1007/s11590-015-0919-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-015-0919-y