Faster Semi-streaming Matchings via Alternating Trees

2024-12-26

AI Summary

This paper studies the $(1+varepsilon)$-approximate maximum matching problem in the semi-streaming model. We propose a novel modeling framework integrating alternating trees with blossom contraction, explicitly incorporating alternating tree structure into the blossom-handling process—thereby simplifying correctness proofs and enabling deterministic greedy augmentation. Our approach reduces the semi-streaming pass complexity from $O(varepsilon^{-19})$ to $O(varepsilon^{-6})$, the first exponential improvement. Moreover, it simultaneously achieves $O(varepsilon^{-6})$ round complexity in both the MPC and CONGEST models. Key technical innovations include: (i) an explicit alternating tree representation; (ii) an abstraction of blossom contraction; (iii) semi-streaming multi-pass scanning; and (iv) generalization of the graph model to support broader applicability. This work establishes the best-known theoretical speedup for distributed and streaming matching algorithms, significantly advancing the state of the art in both models.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Semi-Supervised Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

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.

Problem

• Approximate Maximum Matching
• Computational Speed
• Pairing Process Acceleration

Innovation

• Interlaced Tree Algorithm
• (1+ε)-Approximation
• Maximum Matching Optimization