A flow-dependent quadratic steiner tree problem in the Euclidean plane

We introduce a flow-dependent version of the quadratic Steiner tree problem in the plane. An instance of the problem on a set of embedded sources and a sink asks for a directed tree T spanning these nodes and a bounded number of Steiner points, such that ∑ e∈E(T ) f(e)|e| is a minimum, where f(e) is the flow on edge e. The edges are uncapacitated and the flows are determined additively, i.e., the flow on an edge leaving a node u will be the sum of the flows on all edges entering u. Our motivation for studying this problem is its utility as a model for relay augmentation of wireless sensor networks. In these scenarios one seeks to optimise power consumption – which is predominantly due to communication and, in free space, is proportional to the square of transmission distance – in the network by introducing additional relays. We prove several geometric and combinatorial results on the structure of optimal and locally optimal solution-trees (under various strategies for bounding the number of Steiner points) and describe a geometric linear-time algorithm for constructing such trees with known topologies.