Problem 68: Approximating Rank in the Bounded-Degree Model

Suggested by	Yuichi Yoshida
Source	submitted online
Short link	https://sublinear.info/68

Let $A:\mathbb{F}_p^{m \times n}$ be a matrix such that each row and column has a constant number of non-zero entries (hence, $m = O(n)$). The matrix $A$ is given as a query access: If we specify the $i$-th row, then we obtain the indices $j$ for which $A_{i,j} \neq 0$. Similarly, if we specify the $j$-th column, then we obtain the indices $i$ for which $A_{i,j}\neq 0$. For a parameter $\epsilon > 0$, we want to approximate the rank of $A$ to whithin $\pm \epsilon$. How many queries are needed for this task?

When $p = 2$ and each column has exactly two ones, $A$ can be seen as an incidence matrix, and its rank is equal to $n - c$, where $c$ is the number of connected components. So, we can approximate the rank with $O(1/\epsilon^2)$ queries.

In general, the conjecture is $\Omega(n)$. The difficulty is that not so many techniques are known to show $\Omega(n)$ lower bound for the bounded-degree model. [BogdanovOT-02] shows a lower bound of $\Omega(n)$ for the problem of solving E3LIN2 in the bounded-degree model. However, the lower bound is obtained by considering a distribution of instances of the form $Ax = b$, where $A$ is fixed and $b$ is random. Hence, we cannot directly use the construction for the rank problem as we only have $A$.