A Two-Pass Lower Bound for Semi-Streaming Maximum Matching

TL;DR

This paper establishes a lower bound on the space complexity of two-pass semi-streaming algorithms for maximum matching, linking it to the density of Ruzsa-Szemeredi graphs, and shows that small-constant approximation algorithms are impossible under certain hypotheses.

Contribution

It provides the first lower bound that rules out small-constant approximation algorithms for maximum matching in the semi-streaming model, based on graph density assumptions.

Findings

Lower bound depends on Ruzsa-Szemeredi graph density

Under plausible hypotheses, small-constant approximation algorithms are impossible

Progress on a longstanding open problem in graph streaming literature

Abstract

We prove a lower bound on the space complexity of two-pass semi-streaming algorithms that approximate the maximum matching problem. The lower bound is parameterized by the density of Ruzsa-Szemeredi graphs: * Any two-pass semi-streaming algorithm for maximum matching has approximation ratio at least $(1- \Omega(\frac{\log{RS(n)}}{\log{n}}))$, where $RS(n)$ denotes the maximum number of induced matchings of size $\Theta(n)$ in any $n$-vertex graph, i.e., the largest density of a Ruzsa-Szemeredi graph. Currently, it is known that $n^{\Omega(1/\!\log\log{n})} \leq RS(n) \leq \frac{n}{2^{O(\log^*{\!(n)})}}$ and closing this (large) gap between upper and lower bounds has remained a notoriously difficult problem in combinatorics. Under the plausible hypothesis that $RS(n) = n^{\Omega(1)}$, our lower bound is the first to rule out small-constant approximation two-pass semi-streaming algorithms for the maximum matching problem, making progress on a longstanding open question in the graph streaming literature.