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 A i ⊆ 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{|A i |, n} + n|A|)-time algorithm. Our improvement comes from a technique which we use to reduce each set A i . This technique may also be useful in solving the depot location problem on special classes of graphs, such as trees and planar graphs.
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