Locating a service facility and a rapid transit line

TL;DR

This paper revises a facility and transit line location problem, correcting previous algorithmic errors, providing a clearer solution characterization, and presenting a more efficient algorithm with improved computational complexity.

Contribution

It corrects and simplifies the characterization of optimal solutions and introduces a faster algorithm for the facility and transit line location problem.

Findings

Previous algorithm was incorrect in some cases.

New characterization of optimal solutions provided.

Proposed algorithm runs in O(n^3) time.

Abstract

In this paper we study a facility location problem in the plane in which a single point (facility) and a rapid transit line (highway) are simultaneously located in order to minimize the total travel time of the clients to the facility, using the $L_1$ or Manhattan metric. The rapid transit line is represented by a line segment with fixed length and arbitrary orientation. The highway is an alternative transportation system that can be used by the clients to reduce their travel time to the facility. This problem was introduced by Espejo and Ch\'ia in [7]. They gave both a characterization of the optimal solutions and an algorithm running in $O(n^3\log n)$ time, where $n$ represents the number of clients. In this paper we show that the Espejo and Ch\'ia's algorithm does not always work correctly. At the same time, we provide a proper characterization of the solutions with a simpler proof and give an algorithm solving the problem in $O(n^3)$ time.