Algebraic Algorithms for Linear Matroid Parity Problems

A Algebraic Algorithms for Linear Matroid Parity Problems

We present fast and simple algebraic algorithms for the linear matroid parity problem and its applications. For the linear matroid parity problem, we obtain a simple randomized algorithm with running time O(mrω−1) where m and r are the number of columns and the number of rows and ω ≈ 2.3727 is the matrix multiplication exponent. This improves the O(mrω)-time algorithm by Gabow and Stallmann, an...

New algorithms for linear k-matroid intersection and matroid k-parity problems

We present algorithms for the k-Matroid Intersection Problem and for the Matroid k-Pafity Problem when the matroids are represented over the field of rational numbers and k > 2. The computational complexity of the algorithms is linear in the cardinality and singly exponential in the rank of the matroids. As an application, we describe new polynomially solvable cases of the k-Dimensional Assignm...

Algebraic Algorithms for Matching and Matroid Problems

We present new algebraic approaches for several well-known 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) where n is the number of vertices and ω is the m...

Weighted Linear Matroid Parity

Parallel Complexity for Matroid Intersection and Matroid Parity Problems

Let two linear matroids have the same rank in matroid intersection. A maximum linear matroid intersection (maximum linear matroid parity set) is called a basic matroid intersection (basic matroid parity set), if its size is the rank of the matroid. We present that enumerating all basic matroid intersections (basic matroid parity sets) is in NC, provided that there are polynomial bounded basic m...