Partial inverse min–max spanning tree problem under the weighted bottleneck hamming distance

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∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w^*$$\end{document}, so that F can be extended into a min–max spanning tree with respect to w∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w^*$$\end{document} and the gap between w and w∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$w^*$$\end{document} 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|2log|E|)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(|E|^2\log |E|)$$\end{document}, 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|)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(|E|\log |E|)$$\end{document}. Moreover, we show that these algorithms can be generalized to solve these problems with capacitated constraint.