Sublinear Estimation of Weighted Matchings in Dynamic Data Streams

TL;DR

This paper introduces a new algorithm for estimating maximum weighted matchings in dynamic data streams, achieving constant approximation for certain graph classes and polylogarithmic approximation in random streams, with established space lower bounds.

Contribution

It presents the first constant approximation for maximum matching size in dynamic graph streams for planar and bounded arboricity graphs, extending to weighted matchings, and provides space lower bounds for adversarial streams.

Findings

Constant estimation for planar graphs using $ ilde{O}(n^{4/5})$ space.

Polylogarithmic approximation for general graphs in random order streams.

Space lower bound of $oldsymbol{ ilde{oldsymbol{ ext{O}}}(n^{1-oldsymbol{ ext{O}}(oldsymbol{ ext{varepsilon}})})}$ for adversarial streams.

Abstract

This paper presents an algorithm for estimating the weight of a maximum weighted matching by augmenting any estimation routine for the size of an unweighted matching. The algorithm is implementable in any streaming model including dynamic graph streams. We also give the first constant estimation for the maximum matching size in a dynamic graph stream for planar graphs (or any graph with bounded arboricity) using $\tilde{O}(n^{4/5})$ space which also extends to weighted matching. Using previous results by Kapralov, Khanna, and Sudan (2014) we obtain a $\mathrm{polylog}(n)$ approximation for general graphs using $\mathrm{polylog}(n)$ space in random order streams, respectively. In addition, we give a space lower bound of $\Omega(n^{1-\varepsilon})$ for any randomized algorithm estimating the size of a maximum matching up to a $1+O(\varepsilon)$ factor for adversarial streams.