Tight rank lower bounds for the Sherali–Adams proof system

We consider a proof (more accurately, refutation) system based on the Sherali–Adams ( SA) operator associated with integer linear programming. If F is a CNF contradiction that admits a Resolution refutation of width k and size s , then we prove that the SA rank of F is ≤ k and the SA size of F is ≤ ( k + 1 ) s + 1 . We establish that the SA rank of both the Pigeonhole Principle PHP n − 1 n and the Least Number Principle LNP n is n − 2 . Since the SA refutation system rank-simulates the refutation system of Lovász–Schrijver without semidefinite cuts ( LS), we obtain as a corollary linear rank lower bounds for both of these principles in LS.