Abstract
We extend the techniques developed in Ivanyos et al. (Comput Complex 26(3):717–763, 2017) to obtain a deterministic polynomial-time algorithm for computing the non-commutative rank of linear spaces of matrices over any field. The key new idea that causes a reduction in the time complexity of the algorithm in Ivanyos et al. (2017) from exponential time to polynomial time is a reduction procedure that keeps the blow-up parameter small, and there are two methods to implement this idea: the first one is a greedy argument that removes certain rows and columns, and the second one is an efficient algorithmic version of a result of Derksen & Makam (Adv Math 310:44–63, 2017b), who were the first to observe that the blow-up parameter can be controlled. Both methods rely crucially on the regularity lemma from Ivanyos et al. (2017). In this note, we improve that lemma by removing a coprime condition there.
Highlights
1.1 From the bipartite perfect matching problem to the commutative and non-commutative rank problemsGiven a bipartite graph G = (L ∪ R, E) where |L| = |R|, the celebrated Hall’s marriage theorem states that G has a perfect matching if and only if G has no shrunk subsets: S ⊆ L is called a shrunk subset, if |N (S)| < |S| where N (S) denotes the set of neighbours of S.Consider the following linear algebraic analogue of the bipartite perfect matching problem
Each edge may be viewed as a “partial function” from the left vertex set to the right vertex set
In the linear algebraic setting we shall think of one linear map from U to V as one edge
Summary
1.1 From the bipartite perfect matching problem to the commutative and non-commutative rank problems. While perfect matchings and shrunk subsets are two sides of the same coin for bipartite graphs, non-singular matrices and shrunk subspaces are not in the linear algebraic setting. This gives rise to two natural algorithmic problems. While these two problems are different in general, as suggested by the 3×3 skew-symmetric matrix space, they do coincide in some special cases For this let us recall how the bipartite perfect matching problem and the linear matroid intersection problem can be cast as special instances of both problems. Using Edmonds’ matroid intersection theorem, Lovász showed that in this case the commutative rank and the non-commutative rank of B coincide, and both are equal to the matroid intersection number
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