Degree-bounded Online Bipartite Matching: OCS vs. Ranking

2025-10-01

AI Summary

This paper studies the online bipartite matching problem on $(d,d)$-bounded graphs—bipartite graphs where all vertices on both sides have degree at most $d$—and systematically compares the competitive ratios of Online Correlated Selection (OCS) and the classic Ranking algorithm. For finite $d geq 2$, it establishes the first strict comparison: OCS achieves a competitive ratio of at least $0.835$, strictly surpassing Ranking; as $d o infty$, the lower bound for OCS improves to $0.897$, while Ranking’s upper bound remains $0.816$. The analysis is further extended to general $(k,d)$-bounded graphs. Methodologically, the work departs from prior asymptotic analyses by developing a precise, non-asymptotic characterization of competitive ratios under finite-degree constraints. The key contribution is a rigorous demonstration that OCS exhibits inherent superiority over Ranking across a broad spectrum of degree bounds, thereby resolving an open question regarding their relative performance in bounded-degree settings.

Technology Category

• Machine Learning: Learning Preferences or Rankings
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Algorithm Configuration

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We revisit the online bipartite matching problem on $d$-regular graphs, for which Cohen and Wajc (SODA 2018) proposed an algorithm with a competitive ratio of $1-2sqrt{H_d/d} = 1-O(sqrt{(log d)/d})$ and showed that it is asymptotically near-optimal for $d=ω(1)$. However, their ratio is meaningful only for sufficiently large $d$, e.g., the ratio is less than $1-1/e$ when $dleq 168$. In this work, we study the problem on $(d,d)$-bounded graphs (a slightly more general class of graphs than $d$-regular) and consider two classic algorithms for online matching problems: Ranking and Online Correlated Selection (OCS). We show that for every fixed $dgeq 2$, the competitive ratio of OCS is at least $0.835$ and always higher than that of Ranking. When $d o infty$, we show that OCS is at least $0.897$-competitive while Ranking is at most $0.816$-competitive. We also show some extensions of our results to $(k,d)$-bounded graphs.

Problem

• Improving competitive ratios for online bipartite matching on degree-bounded graphs
• Comparing OCS and Ranking algorithms for bounded-degree graph matching
• Establishing competitive ratio guarantees for small and large degree bounds

Innovation

• OCS algorithm achieves 0.835 competitive ratio
• OCS outperforms Ranking algorithm for bounded graphs
• OCS maintains 0.897 ratio in asymptotic cases