Delay-constrained minimum shortest path trees and related problems

Abstract

Motivated by some applications in communication networks of the diameter-constrained minimum spanning tree (Diamc-MST) problem, we address the delay-constrained minimum shortest path tree (Delayc-MSPT) problem, which is a variation of the Diamc-MST problem. Specifically, given a weighted graph G=(V,E;w,c) and a constant d0, where n=|V|, m=|E|, length function w:E→R+ and cost function c:E→R+, we are asked to determine a minimum shortest path tree among all shortest path trees (in G) whose delays are no more than d0, where the delay of a shortest path tree is the maximum distance (with respect to w(⋅)) from its source to every other leaf in that tree, and the cost of such a shortest path tree is the sum of costs of all edges (with respect to c(⋅)) in that tree. In addition, when a constant d0 is the radius of G, we refer to this version of the Delayc-MSPT problem as the minimum radius minimum shortest path tree (MinRadius-MSPT) problem, and when a constant d0 is the diameter of G, we refer to that version of the Delayc-MSPT problem as the maximum delay minimum shortest path tree (MaxDelay-MSPT) problem. In addition, we consider the diameter-constrained minimum shortest path tree (Diamc-MSPT) problem, except substituting the diameter for the delay of G in the Delayc-MSPT problem, where the diameter of a shortest path tree is the maximum distance (with respect to w(⋅)) between any two leaves in that tree.The main contribution of our paper is to show the following results. (1) We design an exact algorithm to solve the Delayc-MSPT problem and a similar exact algorithm to solve the MinRadius-MSPT problem, respectively, and these two algorithms run in time O(n3); (2) We present an exact algorithm to solve the MaxDelay-MSPT problem, and that algorithm runs in time O(n3); (3) We provide an exact algorithm to solve the special version of the Diamc-MSPT problem, where the constant d0 is exactly the diameter of G, and that algorithm runs in time O(mn2+n3log⁡n).