A new exact algorithm for rectilinear Steiner trees

Given a nite set V of points in the plane (called terminals), the rectilinear Steiner minimal tree is a shortest network of horizontal and vertical lines connecting all the terminals of V. The decision form of this problem has been shown to be NP-complete 10]. The FST concatenation approach has been more successful in practice than all other currently known methods for solving the geometric Steiner tree problems. The nal (and most time-consuming) step in this method is to solve the FST concatenation problem, which is equivalent to nding a minimum spanning tree in a certain hypergraph. We show that the MST in hypergraph problem is NP-complete. We then formulate this problem as an integer program, and present a branch-and-cut algorithm for solving it. We deene STHGP(n) | the spanning tree in hypergraph polytope and prove a number of its properties. In particular, we show that two of the classes of constraints in our formulation deene facets of STHGP(n). We present a complete list of the facets of STHGP(n) for 2 n 4. This algorithm has obtained optimal solutions for all of the OR-library 3] problem instances having 1000 points or fewer. We present detailed computational results on these problems, as well as aggregate results from a more extensive computational study with random data. This represents a substantial advance since the best rectilinear Steiner tree results currently in the literature are 35 points 24], 40 points 18], and 55 points 7]. Using the Euclidean FST generator of Winter and Zachariasen 28], this algorithm has also obtained optimal Euclidean Steiner trees for problems as large as 2000 terminals.

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