Even factors, jump systems, and discrete convexity
Journal of Combinatorial Theory Series B
Computing the maximum degree of minors in mixed polynomial matrices via combinatorial relaxation
IPCO'11 Proceedings of the 15th international conference on Integer programming and combinatoral optimization
On the Kronecker Canonical Form of Mixed Matrix Pencils
SIAM Journal on Matrix Analysis and Applications
A scaling algorithm for maximum weight matching in bipartite graphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Algebraic algorithms for linear matroid parity problems
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Fast matrix rank algorithms and applications
STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
Body-and-cad geometric constraint systems
Computational Geometry: Theory and Applications
A simple reduction from maximum weight matching to maximum cardinality matching
Information Processing Letters
Fast matrix rank algorithms and applications
Journal of the ACM (JACM)
Linear-Time Approximation for Maximum Weight Matching
Journal of the ACM (JACM)
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We present new algebraic approaches for two well-known combinatorial problems: nonbipartite 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 $O(n^\omega)$ where $n$ is the number of vertices and $\omega$ is the matrix multiplication exponent. This resolves the central open problem of Mucha and Sankowski (2004). For matroid intersection, our algorithm has running time $O(nr^{\omega-1})$ for matroids with $n$ elements and rank $r$ that satisfy some natural conditions.