According to D. den Hertog, Interior Point Approach to Linear, Quadratic and Convex Programming, 1994, a linear program with $n$ variables, $n$ constraints and precision $L$ is solvable in $O(n^3L)$ time. Has that been improved upon?
It seems that K.M.Anstreicher has improved the result to $O((n^3/\ln n)L)$ in Anstreicher, Kurt M. "Linear programming in O ([n3/ln n] L) operations." SIAM Journal on Optimization 9, no. 4 (1999): 803-812.. I have not read this paper, but I hope that this answer will help you in some extent.