Efficient Theoretic and Practical Algorithms for Linear Matroid Intersection Problems

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...

Efficient Algorithms for a Family of Matroid Intersection Problems

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...

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...

Valuated Matroid Intersection II: Algorithms

Based on the optimality criteria established in Part I, we show a primal-type cyclecanceling algorithm and a primal-dual-type augmenting algorithm for the valuated independent assignment problem: Given a bipartite graph G = (V , V −;A) with arc weight w : A → R and matroid valuations ω and ω− on V + and V − respectively, find a matching M(⊆ A) that maximizes ∑ {w(a) | a ∈ M}+ ω(∂M) + ω−(∂−M), w...