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 matroid intersections (basic matroid parity sets). For the graphic matroids, We show that constructing all basic matroid intersections is in NC if the number of basic graphic matroid intersections is polynomial bounded. To our knowledge, these algorithms are the first deterministic NC-algorithms for matroid intersection and matroid parity. Our result also answers a question of Harvey [8].