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(n2 log n) time algorithm.
Finally, we show some numerical experiments to verify the efficiency of the algorithms.