An algebraic algorithm for weighted linear matroid intersection

Nicholas J. A. Harvey

Abstract: 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-1W1+ε) 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 recent work of Sankowski (2006) for Weighted Bipartite Matching and Harvey (2006) for Unweighted Linear Matroid Intersection.