The Bounded Path Tree Problem

Abstract

The subject of this paper is the bounded path tree (BPT) problem: An undirected graph $G ( V,E )$ is given whose edges have nonnegative lengths; two subsets I and J of V are also given, and nonnegative constants $U_i $, $W_i $ are associated with each $i \in I$, $j \in J$. The BPT problem asks for a tree of G whose vertex set contains $I \cup J$ and whose path joining vertices i and j is not longer than $U_i + W_j $, for each $i \in I$, $j \in J$. This problem generalizes the shortest path and the minimum longest path spanning tree problem. It complements standard min–max location problems, as it asks for a tree given the facility locations, instead of locating facilities in a given network. In this paper we propose some applications of the BPT problem for the design of emergency and communication networks, show its equivalence to an extension of the absolute center location problem and give an algorithm for its solution. This algorithm requires time $O( k| E | + k | V | \log k )$, where $k = | I \cup J |$, plus time for finding in G all shortest path lengths between a vertex in $I \cup J$ and a vertex in V. We also consider a few simple extensions of the BPT problem, such as those admitting negative or multiple edge lengths, lower (as well as upper) bounds to path lengths, constants $Z_{ij} $ instead of $U_i + W_i $. We show that all these extensions are NP-complete.