An Initial Study of Budgeted Steiner Networks

Abstract

Given a set of terminals, the network with the shortest total length connecting all terminals is a Steiner tree. At the other extreme, with a sufficient total length budget, every terminal can be connected to every other terminal via a straight line, yielding a complete graph over all terminals that connects every terminal pair with a shortest path. In this work, we study a generalization of Steiner trees, asking what happens between these two extremes. For a given total length budget, we seek a network structure that minimizes the sum of the weighted distances between pairs of terminals. Focusing on three terminals with equal pairwise path weights, we characterize the full evolution pathway between the Steiner tree and the complete graph, which contains interesting intermediate structures. Our initial study of such structures, which we call budgeted Steiner networks, opens up many interesting directions for further investigation.