# Breaking the Quadratic Barrier for Matroid Intersection

Publication
In STOC 2021

The matroid intersection problem is a fundamental problem that has been extensively studied for half a century. In the classic version of this problem, we are given two matroids ${\cal M}_1 = (V, {\cal I}_1)$ and ${\cal M}_2 = (V, {\cal I}_2)$ on a comment ground set $V$ of $n$ elements, and then we have to find the largest common independent set $S \in {\cal I}_1 \cap {\cal I}_2$ by making independence oracle queries of the form ''Is $S \in {\cal I}_1$?'' or ''Is $S \in {\cal I}_2$?'' for $S \subseteq V$. The goal is to minimize the number of queries.

Beating the existing $\tilde O(n^2)$ bound, known as the quadratic barrier, is an open problem that captures the limits of techniques from two lines of work. The first one is the classic Cunningham's algorithm [SICOMP 1986], whose $\tilde O(n^2)$-query implementations were shown by CLS+ [FOCS 2019] and Nguyen [2019]. More generally, these algorithms take $\tilde O(nr)$ queries where $r$ denotes the rank which can be as big as $n$. The other one is the general cutting plane method of Lee, Sidford, and Wong [FOCS 2015]. The only progress towards breaking the quadratic barrier requires either {\em approximation} algorithms or a more powerful rank oracle query [CLS+ FOCS 2019]. No exact algorithm with $o(n^2)$ independence queries was known.

In this work, we break the quadratic barrier with a randomized algorithm guaranteeing $\tilde O(n^{9/5})$ independence queries with high probability, and a deterministic algorithm guaranteeing $\tilde O(n^{11/6})$ independence queries. Our key insight is simple and fast algorithms to solve a graph reachability problem that arose in the standard augmenting path framework [Edmonds 1968]. Combining this with previous exact and approximation algorithms leads to our results.