  Various first order approaches have been proposed in the literature to solve
Linear Programming (LP) problems, recently leading to practically efficient
solvers for large-scale LPs. From a theoretical perspective, linear convergence
rates have been established for first order LP algorithms, despite the fact
that the underlying formulations are not strongly convex. However, the
convergence rate typically depends on the Hoffman constant of a large matrix
that contains the constraint matrix, as well as the right hand side, cost, and
capacity vectors.
  We introduce a first order approach for LP optimization with a convergence
rate depending polynomially on the circuit imbalance measure, which is a
geometric parameter of the constraint matrix, and depending logarithmically on
the right hand side, capacity, and cost vectors. This provides much stronger
convergence guarantees. For example, if the constraint matrix is totally
unimodular, we obtain polynomial-time algorithms, whereas the convergence
guarantees for approaches based on primal-dual formulations may have
arbitrarily slow convergence rates for this class. Our approach is based on a
fast gradient method due to Necoara, Nesterov, and Glineur (Math. Prog. 2019);
this algorithm is called repeatedly in a framework that gradually fixes
variables to the boundary. This technique is based on a new approximate version
of Tardos's method, that was used to obtain a strongly polynomial algorithm for
combinatorial LPs (Oper. Res. 1986).
