A Unified Framework for Analysis of Randomized Greedy Matching Algorithms

2026-03-31

AI Summary

This work addresses the lack of a unified analytical framework and tight approximation ratios for stochastic greedy algorithms in graph matching. The authors propose a general analytical paradigm that systematically characterizes the approximation performance of vertex-iterative randomized greedy algorithms by modeling the joint distribution of vertex processing order and preference order. Leveraging probabilistic methods, coupling techniques, and structural graph theory, they establish improved approximation ratios: 0.560 for Ranking and 0.539 for FRanking on general graphs—both state-of-the-art results. Further refinements are achieved on restricted graph classes: a ratio of 0.570 for graphs excluding triangles and pentagons, and an enhanced bound of 0.615 when the shortest odd cycle has length at least 129.

Abstract

Randomized greedy algorithms form one of the simplest yet most effective approaches for computing approximate matchings in graphs. In this paper, we focus on the class of vertex-iterative (VI) randomized greedy matching algorithms, which process the vertices of a graph $G=(V,E)$ in some order $π$ and, for each vertex $v$, greedily match it to the first available neighbor according to a preference order $σ(v)$. Various VI algorithms have been studied, each corresponding to a different distribution over $π$ and $σ(v)$. We develop a unified framework for analyzing this family of algorithms and use it to obtain improved approximation ratios for Ranking and FRanking, the state-of-the-art randomized greedy algorithms for the random-order and adversarial-order settings, respectively. In Ranking, the decision order is drawn uniformly at random and used as the common preference order, whereas FRanking uses an adversarial decision order and a uniformly random preference order shared by all vertices. We obtain an approximation ratio of $0.560$ for Ranking, improving on the $0.5469$ bound of Derakhshan et al. [SODA 2026]. For FRanking, we obtain a ratio of $0.539$, improving on the $0.521$ bound of Huang et al. [JACM 2020]. These results also imply state-of-the-art approximation ratios for oblivious matching and fully online matching problems on general graphs. Our analysis framework also enables us to prove improved approximation ratios for graphs with no short odd cycles. Such graphs form an intermediate class between general graphs and bipartite graphs. In particular, we show that Ranking is at least $0.570$-competitive for graphs that are both triangle-free and pentagon-free. For graphs whose shortest odd cycle has length at least $129$, we prove that Ranking is at least $0.615$-competitive.

Problem

• randomized greedy matching
• approximation ratio
• vertex-iterative algorithms
• Ranking
• graph matching

Innovation

• randomized greedy matching
• unified analysis framework
• approximation ratio
• Ranking algorithm
• odd-cycle-free graphs