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

TL;DR

This paper introduces an improved online algorithm for node-weighted Steiner forest problems, achieving near-optimal competitiveness and extending to more general settings, with significant technical innovations in adversarial models.

Contribution

It presents a nearly optimal randomized algorithm for online node-weighted Steiner forest and extends to prize-collecting problems, introducing a new semi-adaptive adversarial model.

Findings

Achieves $O( ext{log} k ext{log} n)$-competitiveness, improving previous bounds.

Extends results to prize-collecting settings with poly-logarithmic improvements.

Provides the first deterministic algorithm for non-metric facility location with $O( ext{log} |C| ext{log} |F|)$-competitiveness.

Abstract

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 $O(\log |C| \log |F|)$-competitive algorithm for non-metric facility location.