Monotone Circuit Complexity of Matching

TL;DR

This paper proves that the perfect matching function requires exponentially large monotone circuits, significantly improving previous lower bounds, using a novel sunflower lemma for matchings and the approximation method.

Contribution

It introduces a new sunflower lemma for matchings and improves the lower bound on monotone circuit complexity of the perfect matching function.

Findings

Monotone circuit size for perfect matching is at least exponential in n.

The new sunflower lemma for matchings is a key technical tool.

The lower bound surpasses the previous n^{Ω(log n)} bound.

Abstract

We show that the perfect matching function on $n$-vertex graphs requires monotone circuits of size $\smash{2^{n^{\Omega(1)}}}$. This improves on the $n^{\Omega(\log n)}$ lower bound of Razborov (1985). Our proof uses the standard approximation method together with a new sunflower lemma for matchings.