The Gilbert Arborescence Problem

TL;DR

This paper studies the design of minimum cost flow networks connecting multiple sources to a sink in a normed space, characterizing the structure of Steiner points in such networks, called MGAs.

Contribution

It characterizes the local topology of Steiner points in MGAs, showing they typically have degree 3 under various metrics and cost functions.

Findings

Steiner points in MGAs often have degree 3.

The network topology is a tree for concave increasing cost functions.

Characterization applies to a wide range of metrics and real-world cost functions.

Abstract

We investigate the problem of designing a minimum cost flow network interconnecting n sources and a single sink, each with known locations in a normed space and with associated flow demands. The network may contain any finite number of additional unprescribed nodes from the space; these are known as the Steiner points. For concave increasing cost functions, a minimum cost network of this sort has a tree topology, and hence can be called a Minimum Gilbert Arborescence (MGA). We characterise the local topological structure of Steiner points in MGAs, showing, in particular, that for a wide range of metrics, and for some typical real-world cost-functions, the degree of each Steiner point is 3.