This paper addresses a partial inverse combinatorial optimization problem, called the partial inverse min–max spanning tree problem. For a given weighted graph G and a forest F of the graph, the problem is to modify weights at minimum cost so that a bottleneck (min–max) spanning tree of G contains the forest. In this paper, the modifications are measured by the weighted Manhattan distance. The main contribution is to present two algorithms to solve the problem in polynomial time. This result is considerable because the partial inverse minimum spanning tree problem, which is closely related to this problem, is proved to be NP-hard in the literature. Since both the algorithms have the same worse-case complexity, some computational experiments are reported to compare their running time.

本文解决了部分逆组合优化问题，称为最小逆最大生成树问题。对于给定的加权图G和图的森林F，问题在于以最小成本修改权重，从而使G的瓶颈（最小-最大）生成树包含森林。在本文中，修改是通过加权曼哈顿距离来衡量的。主要的贡献是提出了两种算法来解决多项式时间问题。该结果之所以可观，是因为与该问题密切相关的部分最小逆生成树问题在文献中被证明是NP难的。由于两种算法具有相同的最坏情况复杂度，因此，据报导进行了一些计算实验以比较它们的运行时间。