We present new algebraic approaches for two well-known combinatorial problems: non-bipartite 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.

We present a new algebraic algorithm for the classical problem of weighted matroid intersection. This problem generalizes numerous well-known problems, such as bipartite matching, network flow, etc. Our algorithm has running time Õ(nrω−1 W 1+ɛ) for linear matroids with n elements and rank r, where ω is the matrix multiplication exponent, and W denotes the maximum weight of any element. This algorithm is the fastest known when W is small. Our approach builds on the

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To avoid exponential explosion, program verifiers turn the program into a passive form before generating verification conditions. A little known fact is that the passive form makes it easy to use a strongest postcondition calculus to derive the verification condition. In the first part of this paper, the passivation phase is defined precisely enough to allow a study of its algorithmic properties. In the second part, the weakest precondition and strongest postcondition methods are presented in a unified way and then compared empirically.