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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© 2012 Springer-Verlag Berlin Heidelberg
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Wang, BF., Ye, JH., Chen, PJ. (2012). On the Round-Trip 1-Center and 1-Median Problems. In: Rahman, M.S., Nakano, Si. (eds) WALCOM: Algorithms and Computation. WALCOM 2012. Lecture Notes in Computer Science, vol 7157. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28076-4_12
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DOI: https://doi.org/10.1007/978-3-642-28076-4_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-28075-7
Online ISBN: 978-3-642-28076-4
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