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

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(VEcw). 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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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