Editing Open Problems:68
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{{Header | {{Header | ||
| − | |source= | + | |title=Approximating Rank in the Bounded-Degree Model |
| + | |source=online | ||
|who=Yuichi Yoshida | |who=Yuichi Yoshida | ||
}} | }} | ||
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, $m = O(n)$). | 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 $A$ | + | The matrix $A$ is given as a query access: |
If we specify the $i$-th row, then we obtain the indices $j$ for which $A_{i,j} \neq 0$. | 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 $A_{i,j}\neq 0$. | 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 | + | For a parameter $\epsilon > 0$, we want to approximate the rank of $A$ to whithin $\pm \epsilon$. |
| − | How many queries are needed | + | How many queries are needed for this task? |
| − | When $p = 2$ and each | + | 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. | 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, | + | In general, the conjecture is $\Omega(n)$. |
| − | The difficulty | + | The difficulty is that not so many techniques are known to show $\Omega(n)$ lower bound for the bounded-degree model. |
| − | + | {{cite|BogdanovOT-02}} shows a lower bound of $\Omega(n)$ for the problem of testing the satisfiability of E3LIN2 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. | 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 | + | Hence, we cannot directly use the construction for the rank problem as we only have $A$. |
