I have a linear program with $n$ variables, $m$ constraints and $O(nm)$ bit total length (the constraint matrix contains only zeros and ones). I am interested in finding a polynomial time algorithm for exact solving of the linear program, where the dependency on $m$ is as small as possible. More formally:
The time complexity of solving the linear program is known to be $O(n^a m^b)$ for some constants $a$ and $b$. What is the best known pair $a, b$ where the value of $b$ is the minimum possible?