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

This paper addresses the competitiveness of online algorithmsfor two Steiner Tree problems. In the online setting, requests for k ter-minals appear sequentially, and the algorithm must maintain a feasible,incremental solution at all times. In the first problem, the underlyinggraph is directed and has bounded asymmetry, namely the maximumweight of antiparallel links in the graph does not exceed a parameter α.Previous work on this problem has left a gap on the competitive ratiowhich is as large as logarithmic in k in worst case. We present a refinedanalysis, both in terms of the upper and the lower bounds, that closesthe gap and shows that a greedy algorithm is optimal for all values ofthe parameter α.The second part of the paper addresses the Euclidean Steiner tree prob-lem on the plane. Alon and Azar [SoCG 1992, Disc. Comp. Geom. 1993]gave an elegant lower bound on the competitive ratio of any deter-ministic algorithm equal to Ω(log k/ log log k); however, the best (andonly) upper bound known so far is the trivial bound O(log k). We givethe first analysis that makes progress towards closing this long-standinggap. In particular, we present an online algorithm with competitive ratioO(log k/ log log k), provided that the optimal offline Steiner tree belongsin a class of trees with relatively simple structure. This class comprises(among others) not only the adversarial instances of Alon and Azar,but also all rectilinear Steiner trees which can be decomposed in a poly-logarithmic (in k) number of rectilinear full Steiner trees. Interestingly,our analysis is based on techniques developed for the online asymmetricSteiner tree problem.