Maximum matching using linear programming
This approach is described by Grötschel, Lovász and Schrijver in their paper The ellipsoid method and its consequences in combinatorial optimization, as well as in their book Geometric algorithms and combinatorial optimization. Edmonds gave a description of the general matching polytope, which has exponentially many defining linear inequalities. While the simplex method cannot be used to optimize the corresponding linear program, other approaches, such as the ellipsoid algorithm and some interior point methods, can use a separation oracle for efficient optimization (at least theoretically).

There is a third algorithm, due to Micali and V. Vazirani, which is however also quite complicated; see the recent exposition by Vazirani.