Algorithms for the power-p Steiner tree problem in the Euclidean plane

We study the problem of constructing minimum power-p Euclidean k-Steiner trees in the plane. The problem is to find a tree of minimum cost spanning a set of given terminals where, as opposed to the minimum spanning tree problem, at most k additional nodes (Steiner points) may be introduced anywhere in the plane. The cost of an edge is its length to the power of p (where p ≥ 1), and the cost of a network is the sum of all edge costs. We propose two heuristics: a “beaded” minimum spanning tree heuristic; and a heuristic which alternates between minimum spanning tree construction and a local fixed topology minimisation procedure for locating the Steiner points. We show that the performance ratio κ of the beaded-MST heuristic satisfies √ 3 p−1 (1 + 21−p) ≤ κ ≤ 3(2p−1). We then provide two mixed-integer nonlinear programming formulations for the problem, and extend several important geometric properties into valid inequalities. Finally, we combine the valid inequalities with warm-starting and preprocessing to obtain computational improvements for the p = 2 case.