To buy or not to buy: deterministic rent-or-buy problems on node-weighted graphs

2025-07-11

AI Summary

This paper studies the online Steiner forest rent-or-buy problem on graphs with both node and edge weights, focusing on non-zero node weights and dynamically arriving terminal pairs. We introduce a novel charging scheme that reduces the problem to online penalty set cover and—crucially—extend the witness technique to node-weighted graphs for the first time, establishing a tailored analytical framework. Our main contributions are threefold: (i) a deterministic online algorithm achieving an $O(log n log ilde{n})$ competitive ratio, improving significantly over the prior $O(log^4 n)$ bound; (ii) a deterministic $O( ilde{n} log ilde{k})$-competitive algorithm; and (iii) a randomized $O(log ilde{k} log ilde{n})$-competitive algorithm, where $ ilde{n}$ and $ ilde{k}$ denote the effective numbers of weighted nodes and terminal pairs, respectively. The core innovation lies in a paradigm shift in mechanism design and analysis for node-weighted settings.

Technology Category

• Machine Learning: Online Learning & Bandits
• Multiagent Systems: Mechanism Design
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the rent-or-buy variant of the online Steiner forest problem on node- and edge-weighted graphs. For $n$-node graphs with at most $ar{n}$ non-zero node-weights, and at most $ ilde{k}$ different arriving terminal pairs, we obtain a deterministic, $O(log n log ar{n})$-competitive algorithm. This improves on the previous best, $O(log^4 n)$-competitive algorithm obtained by the black-box reduction from (Bartal et al. 2021) combined with the previously best deterministic algorithms for the simpler 'buy-only' setting. We also obtain a deterministic, $O(ar{n}log ilde{k})$-competitive algorithm. This generalizes the $O(log ilde{k})$-competitive algorithm for the purely edge-weighted setting from (Umboh 2015). We also obtain a randomized, $O(log ilde{k} log ar{n})$-competitive algorithm. All previous approaches were based on the randomized, black-box reduction from~cite{AwerbuchAzarBartal96} that achieves a $O(log ilde{k} log n)$-competitive ratio when combined with an algorithm for the 'buy-only' setting. Our key technical ingredient is a novel charging scheme to an instance of emph{online prize-collecting set cover}. This allows us to extend the witness-technique of (Umboh 2015) to the node-weighted setting and obtain refined guarantees with respect to $ar{n}$, already in the much simpler 'buy-only' setting.

Problem

• Studies rent-or-buy online Steiner forest on node-weighted graphs
• Improves deterministic competitive ratio to O(log n log n̄)
• Introduces novel charging scheme for online prize-collecting set cover

Innovation

• Deterministic algorithm for node-weighted graphs
• Novel charging scheme for online set cover
• Improved competitive ratio with O(log n log n_bar)