Vertex quickest 1-center location problem on trees and its inverse problem under weighted l_∞ norm

In view of some shortcomings of traditional vertex 1-center (V1C), we introduce a vertex quickest 1-center (VQ1C) problem on a tree, which aims to find a vertex such that the maximum transmission time to transmit σ units data is minimum. We first characterize some intrinsic properties of VQ1C and design a binary search algorithm in O(n log n) time based on the relationship between V1C and VQ1C, where n is the number of vertices. Furthermore, we investigate the inverse VQ1C problem under weighted l_∞ norm, in which we modify a given capacity vector in an optimal way such that a prespecified vertex becomes the vertex quickest 1-center. We introduce a concept of an effective modification and provide some optimality conditions for the problem. Then we propose an O(n^2 log n) time algorithm. Finally, we show some numerical experiments to verify the efficiency of the algorithms.