Maximum matchings via Gaussian elimination

We present randomized algorithms for finding maximum matchings in general and bipartite graphs. Both algorithms have running time O(n^w), where w is the exponent of the best known matrix multiplication algorithm. Since w < 2.38, these algorithms break through the O(n^2.5) barrier for the matching problem. They both have a very simple implementation in time O(n³) and the only non-trivial element of the O(n^w) bipartite matching algorithm is the fast matrix multiplication algorithm. Our results resolve a long-standing open question of whether Lovasz´s randomized technique of testing graphs for perfect matching in time O(n^w) can be extended to an algorithm that actually constructs a perfect matching.