## Abstract

The inverse max + sum spanning tree (IMSST) problem is studied, which is the first inverse problem on optimization problems with combined minmax–minsum objective functions. Given an edge-weighted undirected network \(G(V,E,c,w)\), the MSST problem is to find a spanning tree \(T\) which minimizes the combined weight \(\max _{e\in T}w(e)+\sum _{e\in T}c(e)\), which can be solved in \(O(m\log n)\) time, where \(m:=|E|\) and \(n:=|V|\). Whereas, in an IMSST problem, a spanning tree \(T_0\) of \(G\) is given, which is not an optimal MSST. A new sum-cost vector \(\bar{c}\) is to be identified so that \(T_0\) becomes an optimal MSST of the network \(G(V,E,\bar{c},w)\), where \(0\le c-l\le \bar{c} \le c+u\) and \(l,u\ge 0\). The objective is to minimize the cost \(\max _{e\in E}q(e)|\bar{c}(e)-c(e)|\) incurred by modifying the sum-cost vector \(c\) under weighted \(l_\infty \) norm, where \(q(e)\ge 1\). We show that the unbounded IMSST problem is a linear fractional combinatorial optimization (LFCO) problem and develop a discrete type Newton method to solve it. Furthermore, we prove an \(O(m)\) bound on the number of iterations, although most LFCO problems can be solved in \(O(m^2 \log m)\) iterations. Therefore, both the unbounded and bounded IMSST problems can be solved by solving \(O(m)\) MSST problems. Computational results show that the algorithms can efficiently solve the problems.

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## Acknowledgments

Research is supported by National Natural Science Foundation of China (10801031) and the State Scholarship Fund of China (201208320355).

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Guan, X., Pardalos, P.M. & Zuo, X. Inverse Max + Sum spanning tree problem by modifying the sum-cost vector under weighted \(l_\infty \) Norm.
*J Glob Optim* **61**, 165–182 (2015). https://doi.org/10.1007/s10898-014-0140-z

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DOI: https://doi.org/10.1007/s10898-014-0140-z