Optimal Streaming Algorithms for Graph Matching

TL;DR

This paper introduces new parameterized streaming algorithms for graph matching that operate efficiently in both dynamic and insert-only models, removing the need for prior knowledge of maximum matching size.

Contribution

It presents the first algorithms that do not require prior bounds on maximum matching size, achieving optimal space and update time complexities in streaming models.

Findings

Dynamic model algorithm computes maximum-weight k-matching in O(Wk^2) space

Insert-only model algorithm computes maximum-weight k-matching in O(k^2) space

Algorithms do not rely on prior bounds of maximum matching size

Abstract

We present parameterized streaming algorithms for the graph matching problem in both the dynamic and the insert-only models. For the dynamic streaming model, we present a one-pass algorithm that, with high probability, computes a maximum-weight $k$-matching of a weighted graph in $\tilde{O}(Wk^2)$ space and that has $\tilde{O}(1)$ update time, where $W$ is the number of distinct edge weights and the notation $\tilde{O}()$ hides a poly-logarithmic factor in the input size. For the insert-only streaming model, we present a one-pass algorithm that runs in $O(k^2)$ space and has $O(1)$ update time, and that, with high probability, computes a maximum-weight $k$-matching of a weighted graph. The space complexity and the update-time complexity achieved by our algorithms for unweighted $k$-matching in the dynamic model and for weighted $k$-matching in the insert-only model are optimal. A notable contribution of this paper is that the presented algorithms {\it do not} rely on the apriori knowledge/promise that the cardinality of \emph{every} maximum-weight matching of the input graph is upper bounded by the parameter $k$. This promise has been a critical condition in previous works, and lifting it required the development of new tools and techniques.