Pfaffian Pairs and Parities: Counting on Linear Matroid Intersection and Parity Problems

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...

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...

Solving the Linear Matroid Parity Problem as a Sequence of Matroid Intersection Problems

In this paper, we present an O(r n) algorithm for the linear matroid parity problem. Our solution technique is to introduce a modest generalization, the non-simple parity problem, and identify an important subclass of non-simple parity problems called 'easy' parity problems which can be solved as matroid intersection problems. We then show how to solve any linear matroid parity problem parametr...

Efficient Theoretic and Practical Algorithms for Linear Matroid Intersection Problems

Efficient algorithms for the matroid intersection problem, both cardinality and weighted versions, are presented. The algorithm for weighted intersection works by scaling the weights. The cardinality algorithm is a special case, but takes advantage of greater structure. Efficiency of the algorithms is illustrated by several implementations on linear matroids. Consider a linear matroid with m el...

Weighted Linear Matroid Parity