On Regularity Lemma and Barriers in Streaming and Dynamic Matching

TL;DR

This paper introduces new algorithms for approximate matchings in dense graphs using the Regularity Lemma, achieving near-optimal results in streaming and dynamic models with sublinear space and update time.

Contribution

It presents the first single-pass streaming algorithm surpassing 1/2-approximation and a dynamic algorithm with near-linear approximation and sublinear update time, both leveraging the Regularity Lemma.

Findings

First single-pass streaming algorithm with >1/2-approximation in sublinear space.

Dynamic algorithm maintaining near-perfect matching with o(n) update time.

Use of Ruzsa-Szemerédi graphs to bound streaming space, novel in algorithm design.

Abstract

We present a new approach for finding matchings in dense graphs by building on Szemer\'edi's celebrated Regularity Lemma. This allows us to obtain non-trivial albeit slight improvements over longstanding bounds for matchings in streaming and dynamic graphs. In particular, we establish the following results for $n$-vertex graphs: * A deterministic single-pass streaming algorithm that finds a $(1-o(1))$-approximate matching in $o(n^2)$ bits of space. This constitutes the first single-pass algorithm for this problem in sublinear space that improves over the $\frac{1}{2}$-approximation of the greedy algorithm. * A randomized fully dynamic algorithm that with high probability maintains a $(1-o(1))$-approximate matching in $o(n)$ worst-case update time per each edge insertion or deletion. The algorithm works even against an adaptive adversary. This is the first $o(n)$ update-time dynamic algorithm with approximation guarantee arbitrarily close to one. Given the use of regularity lemma, the improvement obtained by our algorithms over trivial bounds is only by some $(\log^*{n})^{\Theta(1)}$ factor. Nevertheless, in each case, they show that the ``right'' answer to the problem is not what is dictated by the previous bounds. Finally, in the streaming model, we also present a randomized $(1-o(1))$-approximation algorithm whose space can be upper bounded by the density of certain Ruzsa-Szemer\'edi (RS) graphs. While RS graphs by now have been used extensively to prove streaming lower bounds, ours is the first to use them as an upper bound tool for designing improved streaming algorithms.