On the Simplex Method for 0/1-Polytopes

We present three new pivot rules for the Simplex method for Linear Programs over 0/1-polytopes. We show that the number of nondegenerate steps taken using these three rules is strongly polynomial, linear in the number of variables, and linear in the dimension. Our bounds on the number of steps are asymptotically optimal on several well-known combinatorial polytopes. Our analysis is based on the geometry of 0/1-polytopes and novel modifications to the classical steepest-edge and shadow-vertex pivot rules. We draw interesting connections between our pivot rules and other well-known algorithms in combinatorial optimization. Funding: A. E. Black and J. A. De Loera are grateful for the support received through the National Science Foundation Grants DMS-1818969 and NSF GRFP. L. Sanita is grateful for the support received from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek Grant VI.Vidi.193.087.