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

Junhua Jia, Xiucui Guan (), Qiao Zhang, Xinqiang Qian and Panos M. Pardalos
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Junhua Jia: Southeast University
Xiucui Guan: Southeast University
Qiao Zhang: Southeast University
Xinqiang Qian: Southeast University
Panos M. Pardalos: University of Florida

Journal of Global Optimization, 2022, vol. 84, issue 3, No 7, 715-738

Abstract: Abstract The max+sum spanning tree (MSST) problem is to determine a spanning tree T whose combined weight $$\max _{e\in T}w(e)+\sum _{e\in T}c(e)$$ max e ∈ T w ( e ) + ∑ e ∈ T c ( e ) is minimum for a given edge-weighted undirected network G(V, E, c, w). This problem can be solved within $$O(m \log n)$$ 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 $$\bar{w}$$ w ¯ so that a given $$T^0$$ T 0 becomes an optimal MSST of the new network $$G(V,E,c,\bar{w})$$ G ( V , E , c , w ¯ ) . The IMSST problem under weighted $$l_\infty $$ l ∞ norm is to minimize the modification cost $$\max _{e\in E} q(e)|\bar{w}(e)-w(e)|$$ max e ∈ E q ( e ) | 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 $$O(m^2\log n)$$ 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.

Keywords: Inverse max+sum spanning tree; Weighted $$l_\infty $$ l ∞ norm; Binary search method; Strongly polynomial time algorithm (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10898-022-01170-y

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