Faster Semi-streaming Matchings via Alternating Trees

TL;DR

This paper presents a faster deterministic semi-streaming algorithm for approximate maximum matching, reducing the number of passes needed and simplifying the analysis using alternating trees and blossom notation.

Contribution

It introduces a semi-streaming algorithm with significantly fewer passes for (1+ε)-approximate maximum matching, improving upon previous pass complexities and extending to parallel models.

Findings

Reduced pass complexity from O(ε^{-19}) to O(ε^{-6})

Applicable to MPC and CONGEST models with similar speed-ups

Simplified correctness analysis using alternating trees

Abstract

We design a deterministic algorithm for the $(1+\epsilon)$-approximate maximum matching problem. Our primary result demonstrates that this problem can be solved in $O(\epsilon^{-6})$ semi-streaming passes, improving upon the $O(\epsilon^{-19})$ pass-complexity algorithm by [Fischer, Mitrovi\'c, and Uitto, STOC'22]. This contributes substantially toward resolving Open question 2 from [Assadi, SOSA'24]. Leveraging the framework introduced in [FMU'22], our algorithm achieves an analogous round complexity speed-up for computing a $(1+\epsilon)$-approximate maximum matching in both the Massively Parallel Computation (MPC) and CONGEST models. The data structures maintained by our algorithm are formulated using blossom notation and represented through alternating trees. This approach enables a simplified correctness analysis by treating specific components as if operating on bipartite graphs, effectively circumventing certain technical intricacies present in prior work.