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Inverse max+sum spanning tree problem under weighted \(l_{\infty }\) norm by modifying max-weight vector

Journal of Global Optimization (2022)Cite this article

Abstract

The max+sum spanning tree (MSST) problem is to determine a spanning tree T whose combined weight \(\max _{e\in T}w(e)+\sum _{e\in T}c(e)\) is minimum for a given edge-weighted undirected network G(V, E, c, w). This problem can be solved within \(O(m \log n)\) time, where m and n are the numbers of edges and nodes, respectively. An inverse MSST problem (IMSST) aims to determine a new weight vector \(\bar{w}\) so that a given \(T^0\) becomes an optimal MSST of the new network \(G(V,E,c,\bar{w})\). The IMSST problem under weighted \(l_\infty \) norm is to minimize the modification cost \(\max _{e\in E} q(e)|\bar{w}(e)-w(e)|\), where q(e) is a cost modifying the weight w(e). We first provide some optimality conditions of the problem. Then we propose a strongly polynomial time algorithm in \(O(m^2\log n)\) time based on a binary search and a greedy method. There are O(m) iterations and we solve an MSST problem under a new weight in each iteration. Finally, we perform some numerical experiments to show the efficiency of the algorithm.

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Authors and Affiliations

  1. School of Mathematics, Southeast University, Nanjing, 210096, China

    Junhua Jia, Xiucui Guan, Qiao Zhang & Xinqiang Qian

  2. Center for Applied Optimization, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, USA

    Panos M. Pardalos

  3. LATNA, Higher School of Economics, Nizhni Novgorod, Russia

    Panos M. Pardalos

Authors
  1. Junhua Jia
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  2. Xiucui Guan
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  3. Qiao Zhang
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  4. Xinqiang Qian
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  5. Panos M. Pardalos
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Correspondence to Xiucui Guan.

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Research is supported by National Natural Science Foundation of China (11471073).

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Jia, J., Guan, X., Zhang, Q. et al. Inverse max+sum spanning tree problem under weighted \(l_{\infty }\) norm by modifying max-weight vector. J Glob Optim (2022). https://doi.org/10.1007/s10898-022-01170-y

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  • DOI: https://doi.org/10.1007/s10898-022-01170-y

Keywords

  • Inverse max+sum spanning tree
  • Weighted \(l_\infty \)  norm
  • Binary search method
  • Strongly polynomial time algorithm