Fixed gate point location problems

Abstract

Given a metric space with a set of given facilities, location theory asks to place a new facility which minimizes the distances to the given ones. Many results for a variety of problems with norms or metrics as distances are known in the space ${\mathbb {R}}^n$. This paper handles specific location problems in ${\mathbb {R}}^n$ that have been induced from location problems in the so called phylogenetic tree space coming from an application in biology. Some of the location problems in this space may be transformed to ${\mathbb {R}}^n$ and carry interesting properties. In this paper, we only focus on the resulting problems in ${\mathbb {R}}^n$ and we call them fixed gate point problems. The twist is that one is only allowed to traverse between two orthants by going through a fixed gate point, which induces interesting distances. In this paper, these problems are investigated for three different objective functions and solution methods in form of closed formulas and algorithms are given.