Stronger adversaries grow cheaper forests: online node-weighted Steiner problems

We propose a $O(\log k\log n)$-competitive randomized algorithm for online node-weighted Steiner forest. This is essentially optimal and significantly improves over the previous bound of $O({\log }^{2}k\log n)$ by Hajiaghayi et al. [2017]. In fact, our result extends to the more general prize-collecting setting, improving over previous works by a poly-logarithmic factor. Our key technical contribution is a randomized online algorithm for set cover and non-metric facility location in a new adversarial model which we call semi-adaptive adversaries. As a by-product of our techniques, we obtain the first deterministic $O(\log \mathrm{\mid }C\mathrm{\mid }\log \mathrm{\mid }F\mathrm{\mid })$-competitive algorithm for non-metric facility location.