Delay-Constrained Minimum Shortest Path Trees and Related Problems

Abstract

Motivated by applications in communication networks of the diameter-constrained minimum spanning tree problem, we consider the delay-constrained minimum shortest path tree (DcMSPT) problem. Specifically, given a weighted graph $G=(V,E;w,c)$ and a constant $d_0$, where length function $w: E \rightarrow R^+$ and cost function $c: E \rightarrow R^+$, we are asked to find a minimum cost shortest path tree among all shortest path trees (in G) whose delays are no more than $d_0$, where the delay of a shortest path tree is the maximum distance (depending on $w(\cdot )$) from its source to every other leaves in that tree, and the cost of a shortest path tree is the sum of costs of all edges (depending on $c(\cdot )$) in that tree. Particularly, when a constant $d_0$ is exactly the radius of G, we refer to this version of the DcMSPT problem as the minimum radius minimum shortest path tree (MRMSPT) problem. Similarly, the maximum delay minimum shortest path tree (MDMSPT) problem is asked to find a minimum cost shortest path tree among all shortest path trees (in G) whose delays are exactly the diameter of G.

We obtain the following two main results. (1) We design an exact algorithm in time $O(n^3)$ to solve the DcMSPT problem, and we provide the similar algorithm to solve the MRMSPT problem; (2) We present an exact algorithm in time $O(n^3)$ to solve the MDMSPT problem.

This paper is supported by the National Natural Science Foundation of China [Nos. 11861075, 12101593], Project for Innovation Team (Cultivation) of Yunnan Province [No. 202005AE160006], Key Project of Yunnan Provincial Science and Technology Department and Yunnan University [No. 2018FY001014] and Program for Innovative Research Team (in Science and Technology) in Universities of Yunnan Province [C176240111009]. Jianping Li is also supported by Project of Yunling Scholars Training of Yunnan Province.