Two Heuristics for the Steiner Tree Problem

The Steiner tree problem is to nd the tree with minimal Euclidean length spanning a set of xed points in the plane, given the ability to add points (Steiner points). The problem is NP-hard, so polynomial-time heuristics are desired. We present two such heuristics, both of which utilize an eecient method for computing a locally optimal tree with a given topology. The rst systematically inserts Steiner points between edges of the minimal spanning tree meeting at angles less than 120 degrees, performing a local optimization at the end. The second begins by nding the Steiner tree for three of the xed points. Then, at each iteration, it introduces a new xed point to the tree, connecting it to each possible edge by inserting a Steiner point, and minimizes over all connections, performing a local optimization for each. We present a variety of test cases that demonstrate the strengths and weaknesses of both algorithms.