Problem 68: Approximating Rank in the Bounded-Degree Model

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Suggested by Yuichi Yoshida
Source Bertinoro 2014
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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 $A$ can be accessed via the following types of queries. 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 within $\pm \epsilon n$. How many queries are needed to accomplish this task?

When $p = 2$ and each row has exactly two ones, $A$ can be seen as the incidence matrix of a graph, and its rank is equal to $n - c$, where $c$ is the number of connected components. In this case, the rank can be approximated efficiently, with $\tilde O(1/\epsilon^2)$ queries, because we know how to efficiently approximate $c$ [ChazelleRT-05].

In general, we conjecture that $\Omega(n)$ queries are necessary. The difficulty in showing this lower bound arises from the fact that few techniques for proving $\Omega(n)$ lower bounds for the bounded-degree model are known. Bogdanov, Obata, and Trevisan [BogdanovOT-02] show a lower bound of $\Omega(n)$ for the problem of testing the satisfiability of E3LIN-2 instances 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 apply the construction to the rank problem as we only have $A$.