Please note: This seminar will take place in DC 1304 and online.
Akshay Ramachandran, Postdoctoral Researcher
University of Amsterdam and Centrum Wiskunde & Informatica
Given a linear subspace of matrices $\mathcal{A} := \langle A_{1}, …, A_{m} \rangle \subseteq \mathbb{F}^{n \times n}$, Edmond’s problem asks to decide whether there is a matrix in $\mathcal{A}$ of full rank. This can be rephrased as finding the rank of the matrix $A := \sum_{i=1}^{m} x_{i} A_{i}$ over the field of rational functions $\mathbb{F}(x_{1}, …, x_{m})$. This linear algebraic problem turns out to connect to various problems in combinatorial optimization: maximum matching in graphs, linear matroid intersection, and their common generalization linear matroid matching [Lovasz, 1980].
By the Schwarz-Zippel lemma, there is a simple randomized algorithm to answer this question, but derandomizing this result is a major open question that would be an important breakthrough in complexity theory [KI04].
A related variant, known as the non-commutative Edmond’s problem, asks to compute the rank of $A$ over the “free skew field” in which the variables $x$ no longer commute. Surprisingly, recent work has shown that this seemingly more complicated problem can in fact be solved by a deterministic algorithm in polynomial time.
In this talk, we will discuss one of these algorithms and a recent result [Oki, Soma] relating this problem to the Brascamp-Lieb Theorem from functional analysis. In particular, they show that the algorithm for non-commutative rank can be used to optimize over the Brascamp-Lieb polytope for rank 2 instances, which gives the first truly polynomial time algorithm to maximize the fractional relaxation of linear matroid matching [GP08]. This problem can be considered as a natural “higher rank” generalization of optimization over linear matroid polytopes.
To join this seminar on Zoom, please go to https://uwaterloo.zoom.us/j/91061677783. You can also attend in person in DC 1304.
200 University Avenue West
Waterloo, ON N2L 3G1
Canada
