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In this paper we present a new algorithm for solving linear programs thatrequires only $\tilde{O}(\sqrt{rank(A)}L)$ iterations to solve a linear programwith $m$ constraints, $n$ variables, and constraint matrix $A$, and bitcomplexity $L$. Each iteration of our method consists of solving $\tilde{O}(1)$linear systems and additional nearly linear time computation.  Our method improves upon the previous best iteration bound by factor of$\tilde{\Omega}((m/rank(A))^{1/4})$ for methods with polynomial time computableiterations and by $\tilde{\Omega}((m/rank(A))^{1/2})$ for methods which solveat most $\tilde{O}(1)$ linear systems in each iteration. Our method isparallelizable and amenable to linear algebraic techniques for accelerating thelinear system solver. As such, up to polylogarithmic factors we either match orimprove upon the best previous running times in both depth and work fordifferent ratios of $m$ and $rank(A)$.  Moreover, our method matches up to polylogarithmic factors a theoreticallimit established by Nesterov and Nemirovski in 1994 regarding the use of a"universal barrier" for interior point methods, thereby resolving along-standing open question regarding the running time of polynomial timeinterior point methods for linear programming.

Path Finding I :Solving Linear Programs with \~O(sqrt(rank)) Linear System Solves