Editing Open Problems:68
Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.
The edit can be undone.
Please check the comparison below to verify that this is what you want to do, and then save the changes below to finish undoing the edit.
Latest revision | Your text | ||
Line 1: | Line 1: | ||
{{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. | ||
− | + | Hence, we can approximate the rank with $O(1/\epsilon)$ 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 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. | 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 $A$. |