Beating Two-Thirds For Random-Order Streaming Matching

TL;DR

This paper demonstrates that in the random-order semi-streaming model, it is possible to find a matching approximation better than two-thirds of the maximum, breaking previous theoretical limits.

Contribution

The authors prove that a $(2/3 + ext{constant})$-approximate maximum matching can be achieved in this setting, resolving an open problem and surpassing the longstanding boundary.

Findings

Achieved a $(2/3 + ext{constant})$-approximation in semi-streaming model

Used $O(n ext{ poly}( extlog n))$ space with high probability

Resolved an open problem by Bernstein on approximation limits

Abstract

We study the maximum matching problem in the random-order semi-streaming setting. In this problem, the edges of an arbitrary $n$-vertex graph $G=(V, E)$ arrive in a stream one by one and in a random order. The goal is to have a single pass over the stream, use $n \cdot poly(\log n)$ space, and output a large matching of $G$. We prove that for an absolute constant $\epsilon_0 > 0$, one can find a $(2/3 + \epsilon_0)$-approximate maximum matching of $G$ using $O(n \log n)$ space with high probability. This breaks the natural boundary of $2/3$ for this problem prevalent in the prior work and resolves an open problem of Bernstein [ICALP'20] on whether a $(2/3 + \Omega(1))$-approximation is achievable.