A Near-Tight Bound for the Online Steiner Tree Problem in Graphs of Bounded Asymmetry

Abstract

The edge asymmetry of a directed, edge-weighted graph is defined as the maximum ratio of the weight of antiparallel edges in the graph, and can be used as a measure of the heterogeneity of links in a data communication network. In this paper we provide a near-tight upper bound on the competitive ratio of the Online Steiner Tree problem in graphs of bounded edge asymmetry α. This problem has applications in efficient multicasting over networks with non-symmetric links. We show an improved upper bound of $O \left (\min \left \{ \max \left \{ \alpha \frac{\log k}{\log \alpha}, \alpha \frac{\log k}{\log \log k} \right \} ,k \right \} \right )$ on the competitive ratio of a simple greedy algorithm, for any request sequence of k terminals. The result almost matches the lower bound of $\Omega \left (\min \left \{ \max \left \{ \alpha \frac{\log k}{\log \alpha}, \alpha \frac{\log k}{\log \log k} \right \}, k^{1-\epsilon} \right \} \right )$ (where ε is an arbitrarily small constant) due to Faloutsos et al. [8] and Angelopoulos [2].