Near-Optimal Online Algorithms for Prize-Collecting Steiner Problems

In this paper, we give the first online algorithms with a poly-logarithmic competitive ratio for the node-weighted prize-collecting Steiner tree and Steiner forest problems. The competitive ratios are optimal up to logarithmic factors. In fact, we give a generic technique for reducing online prize-collecting Steiner problems to the fractional version of their non-prize-collecting counterparts losing only a logarithmic factor in the competitive ratio. This reduction is agnostic to the cost model (edge-weighted or node-weighted) of the input graph and applies to a wide class of network design problems including Steiner tree, Steiner forest, group Steiner tree, and group Steiner forest. Consequently, we also give the first online algorithms for the edge-weighted prize-collecting group Steiner tree and group Steiner forest problems with a poly-logarithmic competitive ratio, since corresponding fractional guarantees for the non-prize-collecting variants of these problems were previously known. For the most fundamental problem in this class, namely the prize-collecting Steiner tree problem, we further improve our results. For the node-weighted prize-collecting Steiner tree problem, we use the generic reduction but improve the best known online Steiner tree result from Naor et al [14] on two counts. We improve the competitive ratio by a logarithmic factor to make it optimal (up to constants), and also give a new dual-fitting analysis showing that the competitive ratio holds against the fractional optimum. This result employs a new technique that we call dual averaging which we hope will be useful for other dual-fitting analyses as well. For the edge-weighted prize-collecting Steiner tree problem, we match the optimal (up to constants) competitive ratio of O(log n) that was previously achieved by Qian and Williamson [15] but provide a substantially simpler analysis.