The Gilbert Arborescence Problem

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.

19 pages, 7 figures. arXiv admin note: text overlap with arXiv:0903.2124

References in corpus (2)

• The local Steiner problem in finite-dimensional normed spaces
• Vertex degrees of Steiner Minimal Trees in ${\ell }_p^d$ and other smooth Minkowski spaces