Algebraic Algorithms for Matching and Matroid Problems

We present new algebraic approaches for two well-known combinatorial problems: nonbipartite matching and matroid intersection. Our work yields new randomized algorithms that exceed or match the efficiency of existing algorithms. For nonbipartite matching, we obtain a simple, purely algebraic algorithm with running time $O(n^\omega)$ where $n$ is the number of vertices and $\omega$ is the matrix multiplication exponent. This resolves the central open problem of Mucha and Sankowski (2004). For matroid intersection, our algorithm has running time $O(nr^{\omega-1})$ for matroids with $n$ elements and rank $r$ that satisfy some natural conditions.