Online Algorithm for Fractional Matchings with Edge Arrivals in Graphs of Maximum Degree Three

2026-02-07

Workshop on Approximation and Online Algorithms

AI Summary

This study addresses the maximum cardinality fractional matching problem in graphs of maximum degree three under the online edge-arrival model. The authors propose an online algorithm that achieves a tight competitive ratio of $4/(9 - \sqrt{5}) \approx 0.5914$, marking the first such result in this setting. Through rigorous theoretical analysis, they further establish an upper bound of approximately $0.5807$ on the competitive ratio achievable by any online algorithm for integral matchings, thereby demonstrating a provable performance gap between fractional and integral matchings in this graph class. By integrating techniques from online algorithm design, competitive analysis, and matching theory in graph theory, this work delineates the fundamental performance limits of both matching paradigms for graphs with maximum degree three.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We study online algorithms for maximum cardinality matchings with edge arrivals in graphs of low degree. Buchbinder, Segev, and Tkach showed that no online algorithm for maximum cardinality fractional matchings can achieve a competitive ratio larger than $4/(9-\sqrt 5)\approx 0.5914$ even for graphs of maximum degree three. The negative result of Buchbinder et al. holds even when the graph is bipartite and edges are revealed according to vertex arrivals, i.e. once a vertex arrives, all edges are revealed that include the newly arrived vertex and one of the previously arrived vertices. In this work, we complement the negative result of Buchbinder et al. by providing an online algorithm for maximum cardinality fractional matchings with a competitive ratio at least $4/(9-\sqrt 5)\approx 0.5914$ for graphs of maximum degree three. We also demonstrate that no online algorithm for maximum cardinality integral matchings can have the competitive guarantee $0.5807$, establishing a gap between integral and fractional matchings for graphs of maximum degree three. Note that the work of Buchbinder et al. shows that for graphs of maximum degree two, there is no such gap between fractional and integral matchings, because for both of them the best achievable competitive ratio is $2/3$. Also, our results demonstrate that for graphs of maximum degree three best possible competitive ratios for fractional matchings are the same in the vertex arrival and in the edge arrival models.

Problem

• online algorithm
• fractional matching
• competitive ratio
• edge arrivals
• maximum degree three

Innovation

• online algorithm
• fractional matching
• competitive ratio
• edge arrivals
• maximum degree three