## Algebraic structures and algorithms for matching and matroid problems

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Citations: | 10 - 2 self |

### BibTeX

@MISC{Harvey_algebraicstructures,

author = {Nicholas J. A. Harvey},

title = {Algebraic structures and algorithms for matching and matroid problems},

year = {}

}

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### Abstract

We present new algebraic approaches for several wellknown combinatorial problems, including non-bipartite matching, matroid intersection, and some of their generalizations. Our work yields new randomized algorithms that are the most efficient known. For non-bipartite 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. This algorithm is based on new algebraic results characterizing the size of a maximum intersection in contracted matroids. Furthermore, the running time of this algorithm is essentially optimal.