Improved Steiner tree approximation in graphs

The Steiner tree problem in weighted graphs seeks a minimum weight connected subgraph containing a given subset of the vertices (terminals). We present anew polynomial-time heuristic with an approximation ratio approaching 1 + ln3/2 approximate to 1.55, which improves upon the previously best-known approximation algorithm of [10] with performance ratio approximate to 1.59. In quasi-bipartite graphs (i.e., in graphs where all non-terminals are pairwise disjoint), our algorithm achieves an approximation ratio of approximate to 1.28, whereas the previously best method achieves an approximation ratio approaching 1.5 [19]. For complete graphs with edge weights 1 and 2, we shaw that our heuristic has an approximation ratio approaching approximate to 1.28, which improves upon the previously best-known ratio of 4/3 [4]. Our method is considerably simpler and easier to implement than previous approaches. Our techniques can also be used to prove that the Iterated 1-Steiner heuristic [14] achieves an approximation ratio of 1.5 in quasi-bipartite graphs, thus providing the first known non-trivial performance ratio of this well-known method.