Combinatorial Algorithms for the Uniform-Cost Inverse 1-Center Problem on Weighted Trees

Inverse 1-center problem on a network is to modify the edge lengths or vertex weights within certain bounds so that the prespecified vertex becomes an (absolute) 1-center of the perturbed network and the modifying cost is minimized. This paper focuses on the inverse 1-center problem on a weighted tree with uniform cost of edge length modification, a generalization for the analogous problem on an unweighted tree (Alizadeh and Burkard, Discrete Appl. Math. 159, 706–716, 2011). To solve this problem, we first deal with the weighted distance reduction problem on a weighted tree. Then, the weighted distances balancing problem on two rooted trees is introduced and efficiently solved. Combining these two problems, we derive a combinatorial algorithm with complexity of O(n2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(n^{2})$\end{document} to solve the inverse 1-center problem on a weighted tree if there exists no topology change during the edge length modification. Here, n is the number of vertices in the tree. Dropping this condition, the problem is solvable in O(n2c)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(n^{2}\mathbf {c})$\end{document} time, where c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbf {c}$\end{document} is the compressed depth of the tree. Finally, some special cases of the problem with improved complexity, say linear time, are also discussed.