Min–max spanning tree problem is a classical problem in combinatorial optimization. Its purpose is to find a spanning tree to minimize its maximum edge in a given edge weighted graph. Given a connected graph G, an edge weight vector w and a forest F, the partial inverse min–max spanning tree problem (PIMMST) is to find a new weighted vector \(w^*\), so that F can be extended into a min–max spanning tree with respect to \(w^*\) and the gap between w and \(w^*\) is minimized. In this paper, we research PIMMST under the weighted bottleneck Hamming distance. Firstly, we study PIMMST with value of optimal tree restriction, a variant of PIMMST, and show that this problem can be solved in strongly polynomial time. Then, by characterizing the properties of the value of optimal tree, we present first algorithm for PIMMST under the weighted bottleneck Hamming distance with running time \(O(|E|^2\log |E|)\), where E is the edge set of G. Finally, by giving a necessary and sufficient condition to determine the feasible solution of this problem, we present a better algorithm for this problem with running time \(O(|E|\log |E|)\). Moreover, we show that these algorithms can be generalized to solve these problems with capacitated constraint.

最小-最大生成树问题是组合优化中的经典问题。其目的是找到一棵生成树，以最小化给定边加权图中的最大边。给定一个连通图G、一个边权向量w和一个森林F，部分逆最小最大生成树问题（PIMMST）就是找到一个新的加权向量\(w^*\)，使得F可以扩展为相对于\(w^*\) 的最小–最大生成树，并且w和\(w^*\)之间的间隙最小化。在本文中，我们研究了加权瓶颈汉明距离下的PIMMST。首先，我们研究了具有最优树限制值的 PIMMST（PIMMST 的变体），并表明该问题可以在强多项式时间内解决。然后，通过表征最优树值的属性，我们提出了第一个在加权瓶颈汉明距离下的 PIMMST 算法，运行时间为\(O(|E|^2\log |E|)\)，其中E是G的边集。最后，通过给出确定该问题的可行解的充要条件，我们提出了一个运行时间为O(|E|\log |E|)\) 的更好的算法。此外，我们表明这些算法可以推广到解决这些具有能力约束的问题。