WALCOM 2012: WALCOM: Algorithms and Computation pp 100-111

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

• Biing-Feng Wang
• Jhih-Hong Ye
• Pei-Jung Chen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7157)

## 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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Biing-Feng Wang
• 1
• Jhih-Hong Ye
• 1
• Pei-Jung Chen
• 1
1. 1.Department of Computer ScienceNational Tsing Hua UniversityHsinchuTaiwan, Republic of China