An algebraic algorithm for weighted linear matroid intersection
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
The independent even factor problem
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Improved distributed approximate matching
Proceedings of the twentieth annual symposium on Parallelism in algorithms and architectures
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Fast algorithms for (max, min)-matrix multiplication and bottleneck shortest paths
SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Strongest postcondition of unstructured programs
Proceedings of the 11th International Workshop on Formal Techniques for Java-like Programs
ESA'07 Proceedings of the 15th annual European conference on Algorithms
Zigzag persistent homology in matrix multiplication time
Proceedings of the twenty-seventh annual symposium on Computational geometry
Efficient algorithms for maximum weight matchings in general graphs with small edge weights
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
Dynamic matchings in convex bipartite graphs
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
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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^\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. 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.