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 8]. A new algorithm is presented that computes provably optimal Steiner trees using the \FST concatenation" approach. In the \FST generation" phase, extensive geometric processing is used to identify a set of full Steiner trees (FSTs). In the subsequent FST concatenation phase, a Steiner minimal tree is then constructed by a nding a minimal spanning subset of the FSTs. This FST concatenation approach has been more eecient in practice than all other methods currently known. In previous work 19], 20] the author used problem decomposition methods and a \dumb" backtrack search to concatenate FSTs, solving problem instances with up to 65 terminals. Most 45 point instances could be solved within one CPU day on a workstation. Other more recent results include Martin and Koch 16] (who solve 40 point problems), and FF ossmeier and Kaufmann 5] (who solve 55 point problems). This paper presents two major improvements to the Salowe-Warme algorithm that improve its performance dramatically. The rst is a reenement to the rectilinear FST generator that reduces its runtime empirically from exponential to O(n 3). More importantly, the FST concate-nation problem has been formulated as an integer program that is solved via branch-and-cut. Together these innovations have resulted in provably optimal rectilinear Steiner trees for problems with up to 1000 terminals. Using the Euclidean FST generator of Winter and Zachariasen 23], this branch-and-cut procedure has also obtained optimal Euclidean Steiner trees for problems as large as 2000 terminals.