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English

45 Minutes

Saarbrücken

We consider the problem of commutative rank computation of a given matrix space, $\mathcal{B}\subseteq\mathbb{F}^{n\times n}$. The roblem is fundamental, as it generalizes several computational problems from algebra and combinatorics. For instance, checking if the commutative rank of the space is $n$, subsumes problems such as testing perfect matching in graphs and identity testing of algebraic branching programs. An efficient deterministic computation of the commutative rank is a major open problem, although there is a simple and efficient randomized algorithm for it. Recently, there has been a series of results on computing the non-commutative rank of matrix spaces in deterministic polynomial time. Since the non-commutative rank of any matrix space is at most twice the commutative rank, one immediately gets a deterministic $\frac{1}{2}$-approximation algorithm for the computation of the commutative rank. This leads to a natural question of whether this approximation ratio can be improved.

In this work, we answer this question affirmatively. We present a deterministic $\textrm{PTAS}$ for computing the commutative rank of a given matrix space. More specifically, given a matrix space $\mathcal{B}\subseteq\F^{n\times n}$ and a rational number $\epsilon > 0$, we give an algorithm, that runs in time $O(n^{4+\frac{3}{\epsilon}})$ and computes a matrix $A\in\mathcal{B}$ such that the rank of $A$ is at least $(1-\epsilon)$ times the commutative rank of $\mathcal{B}$. The algorithm is the natural greedy algorithm. It always takes the first set of $k$ matrices that will increase the rank of the matrix constructed so far until it does not find any improvement, where the size of the set $k$ depends on $\epsilon$.
This is joint work with Markus Bläser and Anurag Pandey.

Gorav Jindal

015780883476

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Matrix Spaces, Commutative Rank, PTAS.