Deterministic $(1+\varepsilon)$-Approximate Maximum Matching with $\mathsf{poly}(1/\varepsilon)$ Passes in the Semi-Streaming Model and Beyond

TL;DR

This paper introduces a deterministic semi-streaming algorithm for approximate maximum matching that significantly reduces the number of passes needed, breaking previous exponential barriers and extending its approach to other computational models.

Contribution

It presents the first deterministic polynomial-pass semi-streaming algorithm for (1+ε)-approximate maximum matching, and develops a versatile framework applicable to multiple models.

Findings

Exponential improvement over previous randomized algorithms.

Matches the pass complexity of the best deterministic algorithms up to polynomial factors.

Provides a new framework for efficient approximation in various distributed models.

Abstract

We present a deterministic $(1+\varepsilon)$-approximate maximum matching algorithm in $\mathsf{poly} 1/\varepsilon$ passes in the semi-streaming model, solving the long-standing open problem of breaking the exponential barrier in the dependence on $1/\varepsilon$. Our algorithm exponentially improves on the well-known randomized $(1/\varepsilon)^{O(1/\varepsilon)}$-pass algorithm from the seminal work by McGregor~[APPROX05], the recent deterministic algorithm by Tirodkar with the same pass complexity~[FSTTCS18]. Up to polynomial factors in $1/\varepsilon$, our work matches the state-of-the-art deterministic $(\log n / \log \log n) \cdot (1/\varepsilon)$-pass algorithm by Ahn and Guha~[TOPC18], that is allowed a dependence on the number of nodes $n$. Our result also makes progress on the Open Problem 60 at sublinear.info. Moreover, we design a general framework that simulates our approach for the streaming setting in other models of computation. This framework requires access to an algorithm computing an $O(1)$-approximate maximum matching and an algorithm for processing disjoint $(\mathsf{poly} 1 / \varepsilon)$-size connected components. Instantiating our framework in $\mathsf{CONGEST}$ yields a $\mathsf{poly}(\log{n}, 1/\varepsilon)$ round algorithm for computing $(1+\varepsilon$)-approximate maximum matching. In terms of the dependence on $1/\varepsilon$, this result improves exponentially state-of-the-art result by Lotker, Patt-Shamir, and Pettie~[LPSP15]. Our framework leads to the same quality of improvement in the context of the Massively Parallel Computation model as well.