# The Matching Problem in General Graphs is in Quasi-NC

## TL;DR

This paper demonstrates that the perfect matching problem in general graphs can be solved in Quasi-NC, providing a deterministic parallel algorithm by derandomizing the Isolation Lemma, extending previous bipartite graph results.

## Contribution

It introduces a deterministic parallel algorithm for perfect matching in general graphs, extending the derandomization of the Isolation Lemma beyond bipartite graphs.

## Key findings

- Deterministic parallel algorithm for perfect matching in general graphs.
- Runs in O(log^3 n) time with n^{O(log^2 n)} processors.
- Relies on laminar structure of perfect matching polytope faces.

## Abstract

We show that the perfect matching problem in general graphs is in Quasi-NC. That is, we give a deterministic parallel algorithm which runs in $O(\log^3 n)$ time on $n^{O(\log^2 n)}$ processors. The result is obtained by a derandomization of the Isolation Lemma for perfect matchings, which was introduced in the classic paper by Mulmuley, Vazirani and Vazirani [1987] to obtain a Randomized NC algorithm. Our proof extends the framework of Fenner, Gurjar and Thierauf [2016], who proved the analogous result in the special case of bipartite graphs. Compared to that setting, several new ingredients are needed due to the significantly more complex structure of perfect matchings in general graphs. In particular, our proof heavily relies on the laminar structure of the faces of the perfect matching polytope.