On the Round-Trip 1-Center and 1-Median Problems

Abstract

This paper studies the round-trip single-facility location problem, in which a set A of collection depots is given and the service distance of a customer is defined to be the distance from the server, to the customer, then to a depot, and back to the server. (The input is a graph G whose vertices and edges are weighted, whose vertices represent client positions, and that the set of depots is specified in the input as a subset of the points on G.) We consider the restricted version, in which each customer i is associated with a subset Ai⊆A of depots that i can potentially select from and use. Improved algorithms are proposed for the round-trip 1-center and 1-median problems on a general graph. For the 1-center problem, we give an $O(mn \lg n)$ -time algorithm, where n and m are, respectively, the numbers of vertices and edges. For the 1-median problem, we show that the problem can be solved in $O(\min\{mn \lg n, mn + n^{2} \lg n + n|A|\})$ time. In addition, assuming that a matrix that stores the shortest distances between every pair of vertices is given, we give an O(n∑i min {|Ai|, n}+n|A|)-time algorithm. Our improvement comes from a technique which we use to reduce each set Ai. This technique may also be useful in solving the depot location problem on special classes of graphs, such as trees and planar graphs.