Průša and Werner (2019) show that the general linear programming problem reduces in nearly linear time to the LP relaxations of many classical NP-hard problems (assuming sparse encoding of instances). As the authors write:
Arguably, the most important consequences of our reductions are constraints on algorithms to solve the LP relaxations. Leaving runtime aside, they show that such algorithms cannot be arbitrarily simple since they must be able to solve any linear program.
Linear programming is an important tool used to solve integer linear programs (via the LP-based branch and bound approach). There has been a huge progress towards solving such integer programs. However, there does not seem to be much progress in solving the general linear programming problem. As far as I know, the classical simplex algorithm or its dual variant is still used in modern IP solvers (even as the default LP algorithm).
Are there are any new algorithms that could potentially beat the simplex algorithm in practice (at least on average)? If not, then I am wondering why?
The result of Průša and Werner implies that no matter how good the underlying formulation is (or no matter how good the valid inequalities can be), we still need to solve the resulting linear program (i.e., ANY linear program) efficiently to be able to solve large problems.