On Estimating Maximum Matching Size in Graph Streams

TL;DR

This paper investigates the space complexity of estimating maximum matching size in graph streams, providing new upper and lower bounds for both insertion-only and dynamic models, and extending results to matrix rank estimation.

Contribution

It introduces new upper bounds for approximate matching size estimation in streaming models and establishes tight lower bounds, including for dense graphs and matrix rank estimation.

Findings

Upper bounds: $ ilde{O}(n^2/ ext{approximation}^4)$ for dynamic streams, $ ilde{O}(n/ ext{approximation}^2)$ for insertion-only streams.

Lower bounds: $ ilde{ ext{Omega}}(rac{ ext{sqrt}(n)}{ ext{approximation}^{2.5}})$ for dynamic streams, $ ilde{ ext{Omega}}(n/ ext{approximation}^2)$ for dense graphs.

Extension of bounds to matrix rank estimation, providing near-optimal lower bounds for dense matrices.

Abstract

We study the problem of estimating the maximum matching size in graphs whose edges are revealed in a streaming manner. We consider both insertion-only streams and dynamic streams and present new upper and lower bound results for both models. On the upper bound front, we show that an $\alpha$-approximate estimate of the matching size can be computed in dynamic streams using $\widetilde{O}({n^2/\alpha^4})$ space, and in insertion-only streams using $\widetilde{O}(n/\alpha^2)$-space. On the lower bound front, we prove that any $\alpha$-approximation algorithm for estimating matching size in dynamic graph streams requires $\Omega(\sqrt{n}/\alpha^{2.5})$ bits of space, even if the underlying graph is both sparse and has arboricity bounded by $O(\alpha)$. We further improve our lower bound to $\Omega(n/\alpha^2)$ in the case of dense graphs. Furthermore, we prove that a $(1+\epsilon)$-approximation to matching size in insertion-only streams requires RS$(n) \cdot n^{1-O(\epsilon)}$ space; here, RS${n}$ denotes the maximum number of edge-disjoint induced matchings of size $\Theta(n)$ in an $n$-vertex graph. It is a major open problem to determine the value of RS$(n)$, and current results leave open the possibility that RS$(n)$ may be as large as $n/\log n$. We also show how to avoid the dependency on the parameter RS$(n)$ in proving lower bound for dynamic streams and present a near-optimal lower bound of $n^{2-O(\epsilon)}$ for $(1+\epsilon)$-approximation in this model. Using a well-known connection between matching size and matrix rank, all our lower bounds also hold for the problem of estimating matrix rank. In particular our results imply a near-optimal $n^{2-O(\epsilon)}$ bit lower bound for $(1+\epsilon)$-approximation of matrix ranks for dense matrices in dynamic streams, answering an open question of Li and Woodruff (STOC 2016).