On the Power and Limitations of Branch and Cut

The Stabbing Planes proof system [Paul Beame et al., 2018] was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas - certain unsatisfiable systems of linear equations od 2 - which are canonical hard examples for many algebraic proof systems. In a recent (and surprising) result, Dadush and Tiwari [Daniel Dadush and Samarth Tiwari, 2020] showed that these short refutations of the Tseitin formulas could be translated into quasi-polynomial size and depth Cutting Planes proofs, refuting a long-standing conjecture. This translation raises several interesting questions. First, whether all Stabbing Planes proofs can be efficiently simulated by Cutting Planes. This would allow for the substantial analysis done on the Cutting Planes system to be lifted to practical mixed integer programming solvers. Second, whether the quasi-polynomial depth of these proofs is inherent to Cutting Planes.