Lower Bounds on the Sizes of Integer Programs without Additional Variables

For a given set X ⊆ Z of integer points, we investigate the smallest number of facets of any polyhedron whose set of integer points is conv(X) ∩ Z. This quantity, which we call the relaxation complexity of X, corresponds to the smallest number of linear inequalities of any integer program having X as the set of feasible solutions that gets along without auxiliary variables. We show that the use of auxiliary variables is essential for constructing polynomial size integer programming formulations in many relevant cases. In particular, we provide asymptotically tight exponential lower bounds on the relaxation complexity of the integer points of several well-known combinatorial polytopes, including the traveling salesman polytope and the spanning tree polytope.