  In breakthrough work, Tardos (Oper. Res. '86) gave a proximity based
framework for solving linear programming (LP) in time depending only on the
constraint matrix in the bit complexity model. In Tardos's framework, one
reduces solving the LP $\min \langle c,{x}\rangle$, $Ax=b$, $x \geq 0$, $A \in
\mathbb{Z}^{m \times n}$, to solving $O(nm)$ LPs in $A$ having small integer
coefficient objectives and right-hand sides using any exact LP algorithm. This
gives rise to an LP algorithm in time poly$(n,m\log\Delta_A)$, where $\Delta_A$
is the largest subdeterminant of $A$. A significant extension to the real model
of computation was given by Vavasis and Ye (Math. Prog. '96), giving a
specialized interior point method that runs in time poly$(n,m,\log\bar\chi_A)$,
depending on Stewart's $\bar{\chi}_A$, a well-studied condition number.
  In this work, we extend Tardos's original framework to obtain such a running
time dependence. In particular, we replace the exact LP solves with approximate
ones, enabling us to directly leverage the tremendous recent algorithmic
progress for approximate linear programming. More precisely, we show that the
fundamental "accuracy" needed to exactly solve any LP in $A$ is inverse
polynomial in $n$ and $\log\bar{\chi}_A$. Plugging in the recent algorithm of
van den Brand (SODA '20), our method computes an optimal primal and dual
solution using ${O}(m n^{\omega+1} \log (n)\log(\bar{\chi}_A+n))$ arithmetic
operations, outperforming the specialized interior point method of Vavasis and
Ye and its recent improvement by Dadush et al (STOC '20).
  At a technical level, our framework combines together approximate LP
solutions to compute exact ones, making use of constructive proximity theorems
-- which bound the distance between solutions of "nearby" LPs -- to keep the
required accuracy low.
