Open Problems in Sublinear Algorithms - User contributions [en]

Open Problems:68

2014-06-07T06:28:56Z

119.72.199.237:

{{Header
|title=Approximating Rank in the Bounded-Degree Model
|source=online
|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, $m = O(n)$).
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 $\pm \epsilon n$.
How many queries are needed for this task?

When $p = 2$ and each column 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.
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.
{{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.
Hence, we cannot directly apply the construction to the rank problem as we only have $A$.

Open Problems:68

2014-06-07T06:27:43Z

119.72.199.237:

{{Header
|title=Approximating Rank in the Bounded-Degree Model
|source=online
|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, $m = O(n)$).
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 $\pm \epsilon n$.
How many queries are needed for this task?

When $p = 2$ and each column 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.
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.
{{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.
Hence, we cannot directly use the construction for the rank problem as we only have $A$.

Open Problems:68

2014-06-07T06:27:17Z

119.72.199.237:

{{Header
|title=Approximating Rank in the Bounded-Degree Model
|source=online
|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, $m = O(n)$).
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 $\pm \epsilon n$.
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.
{{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.
Hence, we cannot directly use the construction for the rank problem as we only have $A$.