# 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 *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.

## Keywords

Location Problem Planar Graph Distance Matrix Time Algorithm Piecewise Linear Function## Preview

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