Maximum Matching in Turnstile Streams

TL;DR

This paper investigates the space complexity of approximating maximum bipartite matching in turnstile streaming models, establishing tight bounds that highlight the difficulty introduced by edge deletions.

Contribution

It provides the first lower bounds for approximation algorithms in turnstile streams with deletions, showing significant space requirements for certain approximation ratios.

Findings

No $O(n)$ space 2-approximation exists with deletions

Lower bounds match upper bounds up to logarithmic factors

Results are proved in the simultaneous message communication model

Abstract

We consider the unweighted bipartite maximum matching problem in the one-pass turnstile streaming model where the input stream consists of edge insertions and deletions. In the insertion-only model, a one-pass $2$-approximation streaming algorithm can be easily obtained with space $O(n \log n)$, where $n$ denotes the number of vertices of the input graph. We show that no such result is possible if edge deletions are allowed, even if space $O(n^{3/2-\delta})$ is granted, for every $\delta > 0$. Specifically, for every $0 \le \epsilon \le 1$, we show that in the one-pass turnstile streaming model, in order to compute a $O(n^{\epsilon})$-approximation, space $\Omega(n^{3/2 - 4\epsilon})$ is required for constant error randomized algorithms, and, up to logarithmic factors, space $O( n^{2-2\epsilon} )$ is sufficient. Our lower bound result is proved in the simultaneous message model of communication and may be of independent interest.