Abstract
We consider the inverse optimal value problem on minimum spanning tree under unit \(l_{\infty }\) norm. Given an edge weighted connected undirected network \(G=(V, E, {\varvec{w}})\) and a spanning trees \(T^0\), we aim to modify the weights of the edges such that \(T^0\) is the minimum spanning tree under the new weight vector whose weight is equal to a given value K and the modification cost under unit \(l_{\infty }\) norm is minimized. We present a mathematical model of the problem. After analyzing the properties, we propose a sufficient and necessary condition for optimal solutions of the problem. Then we develop a strongly polynomial time algorithm with running time O(|V||E|). Finally, we give an example to demonstrate the algorithm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ahmed, S., Guan, Y.P.: The inverse optimal value problem. Math. Program. 102(1), 91–110 (2005)
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network Flows: Theory, Algorithms, and Applications. Prentice Hall, Englewood Cliffs (1993)
Ahuja, R.K., Orlin, J.B.: A faster algorithm for the inverse spanning tree problem. J. Algorithms 34, 177–193 (2000)
Cai, M.C., Duin, C.W., Yang, X., Zhang, J.: The partial inverse minimum spanning tree problem when weight increase is forbidden. Eur. J. Oper. Res. 188, 348–353 (2008)
Guan, X.C., He, X.Y., Pardalos, P.M., Zhang, B.W.: Inverse max+sum spanning tree problem under hamming distance by modifying the sum-cost vector. J. Glob. Optim. 69(4), 911–925 (2017)
Guan, X.C., Pardalos, P.M., Zhang, B.W.: Inverse max+sum spanning tree problem under weighted \(l_1\) norm by modifying the sum-cost vector. Optim. Lett. 12(5), 1065–1077 (2018)
He, Y., Zhang, B.W., Yao, E.Y.: Weighted inverse minimum spanning tree problems under Hamming distance. J. Comb. Optim. 9(1), 91–100 (2005)
Hochbaum, D.S.: Efficient algorithms for the inverse spanning-tree problem. Oper. Res. 51, 785–797 (2003)
Lai, T., Orlin, J.: The complexity of preprocessing. Research Report of Sloan School of Management, MIT (2003)
Li, S., Zhang, Z., Lai, H.J.: Algorithms for constraint partial inverse matroid problem with weight increase forbidden. Theor. Comput. Sci. 640, 119–124 (2016)
Lv, Y.B., Hua, T.S., Wan, Z.P.: A penalty function method for solving inverse optimal value problem. J. Comput. Appl. Math. 220(1–2), 175–180 (2008)
Liu, L.C., Yao, E.Y.: Inverse min-max spanning tree problem under the weighted sum-type Hamming distance. Theor. Comput. Sci. 396, 28–34 (2008)
Sokkalingam, P.T., Ahuja, R.K., Orlin, J.B.: Solving inverse spanning tree problems through network flow techniques. Oper. Res. 47, 291–298 (1999)
Yang, X.G., Zhang, J.Z.: Some inverse min-max network problems under weighted \(l_1\) and \(l_{\infty }\) norms with bound constraints on changes. J. Comb. Optim. 13(2), 123–135 (2007)
Zhang, B.W., Zhang, J.Z., He, Y.: Constrained inverse minimum spanning tree problems under bottleneck-type Hamming distance. J. Glob. Optim. 34, 467–474 (2006)
Zhang, J.Z., Liu, Z.H., Ma, Z.F.: On the inverse problem of minimum spanning tree with partition constraints. Math. Methods Oper. Res. 44, 171–188 (1996)
Zhang, J.Z., Ma, Z.F.: A network flow method for solving some inverse combinatorial optimization problems. Optimization 37, 59–72 (1996)
Zhang, J.Z., Xu, S.J., Ma, Z.F.: An algorithm for inverse minimum spanning tree problem. Optim. Methods Softw. 8, 69–84 (1997)
Acknowledgements
Research is supported by National Natural Science Foundation of China (11471073) and Chinese Universities Scientific Fund (2018B44014).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhang, B., Guan, X. & Zhang, Q. Inverse optimal value problem on minimum spanning tree under unit \(l_{\infty }\) norm. Optim Lett 14, 2301–2322 (2020). https://doi.org/10.1007/s11590-020-01553-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-020-01553-8