On the Competitiveness of the Online Asymmetric and Euclidean Steiner Tree Problems

This paper addresses the competitiveness of online algorithms for two Steiner Tree problems. In the first problem, the underlying graph is directed and has bounded asymmetry, namely the maximum weight of antiparallel links in the graph does not exceed a parameter α. Previous work on this problem has left a gap on the competitive ratio which is as large as logarithmic in k. We present a refined analysis, both in terms of the upper and the lower bounds, that closes the gap and shows that a greedy algorithm is optimal for all values of the parameter α. The second part of the paper addresses the Euclidean Steiner tree problem on the plane. Alon and Azar [SoCG 1992, Disc. Comp. Geom. 1993] gave an elegant lower bound on the competitive ratio of any deterministic algorithm equal to Ω(logk/ loglogk); however, the best known upper bound is the trivial bound O(logk). We give the first analysis that makes progress towards closing this long-standing gap. In particular, we present an online algorithm with competitive ratio O(logk/ loglogk), provided that the optimal offline Steiner tree belongs in a class of trees with relatively simple structure. This class comprises not only the adversarial instances of Alon and Azar, but also all rectilinear Steiner trees which can be decomposed in a polylogarithmic number of rectilinear full Steiner trees.