The Extension Complexity of Polytopes with Bounded Integral Slack Matrices.

We show that any bounded integral function f:A×B↦{0,1,⋯,Δ} with rank r has deterministic communication complexity ΔO(Δ)·r·logr, where the rank of f is defined to be the rank of the A×B matrix whose entries are the function values. As a corollary, we show that any n-dimensional polytope that admits a slack matrix with entries from {0,1,⋯,Δ} has extension complexity at most exp(ΔO(Δ)·n·logn).