An Asymptotically Optimal Algorithm for Maximum Matching in Dynamic Streams

TL;DR

This paper introduces an asymptotically optimal streaming algorithm for maximum matching in dynamic graphs, matching the best known lower bounds and advancing the understanding of space-efficient graph algorithms.

Contribution

It presents the first algorithm to achieve optimal space complexity for dynamic streaming maximum matching, matching lower bounds up to constant factors, and introduces a novel sketching technique.

Findings

Achieves $O(n^2/ ext{alpha}^3)$ space complexity for approximate maximum matching.

Identifies key 'hard' graph instances for the problem.

Develops a new sketching method that improves space efficiency.

Abstract

We present an algorithm for the maximum matching problem in dynamic (insertion-deletions) streams with *asymptotically optimal* space complexity: for any $n$-vertex graph, our algorithm with high probability outputs an $\alpha$-approximate matching in a single pass using $O(n^2/\alpha^3)$ bits of space. A long line of work on the dynamic streaming matching problem has reduced the gap between space upper and lower bounds first to $n^{o(1)}$ factors [Assadi-Khanna-Li-Yaroslavtsev; SODA 2016] and subsequently to $\text{polylog}{(n)}$ factors [Dark-Konrad; CCC 2020]. Our upper bound now matches the Dark-Konrad lower bound up to $O(1)$ factors, thus completing this research direction. Our approach consists of two main steps: we first (provably) identify a family of graphs, similar to the instances used in prior work to establish the lower bounds for this problem, as the only "hard" instances to focus on. These graphs include an induced subgraph which is both sparse and contains a large matching. We then design a dynamic streaming algorithm for this family of graphs which is more efficient than prior work. The key to this efficiency is a novel sketching method, which bypasses the typical loss of $\text{polylog}{(n)}$-factors in space compared to standard $L_0$-sampling primitives, and can be of independent interest in designing optimal algorithms for other streaming problems.