A Simple $(1-\epsilon)$-Approximation Semi-Streaming Algorithm for Maximum (Weighted) Matching

TL;DR

This paper introduces a simple semi-streaming algorithm that achieves near-optimal approximation for maximum bipartite and weighted matchings in logarithmic passes, using multiplicative weight updates and primal-dual analysis.

Contribution

It presents a straightforward semi-streaming algorithm for near-approximate maximum (weighted) matching, matching state-of-the-art performance with a simpler approach.

Findings

Achieves $(1- ext{epsilon})$-approximation in $O(rac{ ext{log}(n)}{ ext{epsilon}})$ passes.

Uses multiplicative weight update method with primal-dual analysis.

Extends to weighted matchings in general graphs.

Abstract

We present a simple semi-streaming algorithm for $(1-\epsilon)$-approximation of bipartite matching in $O(\log{\!(n)}/\epsilon)$ passes. This matches the performance of state-of-the-art "$\epsilon$-efficient" algorithms -- the ones with much better dependence on $\epsilon$ albeit with some mild dependence on $n$ -- while being considerably simpler. The algorithm relies on a direct application of the multiplicative weight update method with a self-contained primal-dual analysis that can be of independent interest. To show case this, we use the same ideas, alongside standard tools from matching theory, to present an equally simple semi-streaming algorithm for $(1-\epsilon)$-approximation of weighted matchings in general (not necessarily bipartite) graphs, again in $O(\log{\!(n)}/\epsilon)$ passes.