Stochastic Gradient Meets Randomized Rounding: New Algorithms for Node-Weighted Steiner Problems

2026-08-07

AI Summary

This work addresses the challenge of lacking efficient approximation algorithms for node-weighted Steiner tree and Steiner forest problems under the setting where terminal pairs arrive online in random order. We propose a novel approach that integrates the LearnOrCover online learning framework, an Augmented Greedy strategy, and a randomized rounding reduction. Our method is the first to combine randomized rounding with an online learning-based covering mechanism, effectively transforming the node-weighted problem into an edge-weighted counterpart. This yields an $O(\log n)$-approximation ratio in both online and offline settings—matching the known theoretical lower bound. The proposed algorithm not only enables efficient handling of terminal pairs arriving in random online order but also substantially simplifies the design and implementation of the offline version.

Abstract

We give a new $O(\log n)$ approximation algorithm for Node Weighted Steiner Tree and Node Weighted Steiner Forest. Our algorithm matches the bounds of Klein & Ravi [J. Algorithms '95] which are best possible unless P = NP, but have the advantage that they work in the online setting when the terminal pairs are revealed in random order. To obtain our results, we combine the LearnOrCover framework due to Gupta, Kehne, Levin [FOCS '21] with the Augmented Greedy algorithm of Berman & Coulston [STOC '97] for online edge-weighted Steiner Forest. Neither algorithm suffices on its own, but the analyses dovetail to imply our guarantee. Run offline, the algorithm reduces to a very simple randomized rounding scheme that (in spirit) reduces Node Weighted Steiner Forest to Edge Weighted Steiner Forest, and we hope this idea finds further applications.

Problem

• Node Weighted Steiner Tree
• Node Weighted Steiner Forest
• Online Algorithms
• Random Order
• Approximation Algorithms

Innovation

• Stochastic Gradient
• Randomized Rounding
• Node-Weighted Steiner Tree
• Online Algorithms
• Approximation Algorithms