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 ω ) where n is the number of vertices and ω 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 ω-1 ) for matroids with n elements and rank r that satisfy some natural conditions.
['Nicholas J. A. Harvey']
ALGEBRAIC ALGORITHMS FOR MATCHING AND MATROID PROBLEMS