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Linear Programming asks for $x\in\mathbb R^n$ such that $Ax\leq L$ holds where $A\in\mathbb R^{m\times n}$ and $L\in\mathbb R^m$ are given. Karmarkar has shown that $\ell$ is the number of bits of input to the algorithm then we can find $x$ in$O(n^{3.5} \ell)$ operations on $O(\ell)$ digit numbers, as compared to $O(n^6 \ell)$ such operations for the ellipsoid algorithm. The runtime of Karmarkar's algorithm is thus $(n^{3.5} \ell^2 \log \ell \log \log \ell)$ using fast integer multiplication.

  1. What is the best algorithm and its complexity that is currently known and what is believed to be the optimal?

  2. What is the best when $n$ or $m$ is fixed?

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