The Sparse Awakens: Streaming Algorithms for Matching Size Estimation in Sparse Graphs

TL;DR

This paper introduces new streaming algorithms for estimating maximum matching size in sparse graphs, significantly improving space complexity and leveraging streaming order and structural properties.

Contribution

It presents the first one-pass streaming algorithms for maximum matching size estimation in sparse graphs with improved space bounds for both insert-only and dynamic streams.

Findings

One-pass algorithm for insert-only streams with O(c log^2 n) space.

One-pass algorithm for dynamic streams with O~(c^{10/3} n^{2/3}) space.

Improved approximation factors for maximum matching in sparse graphs.

Abstract

Estimating the size of the maximum matching is a canonical problem in graph algorithms, and one that has attracted extensive study over a range of different computational models. We present improved streaming algorithms for approximating the size of maximum matching with sparse (bounded arboricity) graphs. * Insert-Only Streams: We present a one-pass algorithm that takes O(c log^2 n) space and approximates the size of the maximum matching in graphs with arboricity c within a factor of O(c). This improves significantly on the state-of-the-art O~(cn^{2/3})-space streaming algorithms. * Dynamic Streams: Given a dynamic graph stream (i.e., inserts and deletes) of edges of an underlying c-bounded arboricity graph, we present a one-pass algorithm that uses space O~(c^{10/3}n^{2/3}) and returns an O(c)-estimator for the size of the maximum matching. This algorithm improves the state-of-the-art O~(cn^{4/5})-space algorithms, where the O~(.) notation hides logarithmic in $n$ dependencies. In contrast to the previous works, our results take more advantage of the streaming access to the input and characterize the matching size based on the ordering of the edges in the stream in addition to the degree distributions and structural properties of the sparse graphs.