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We consider the Sherali-Adams (SA) refutation system together with theunusual binary encoding of certain combinatorial principles. For the unaryencoding of the Pigeonhole Principle and the Least Number Principle, it isknown that linear rank is required for refutations in SA, although both admitrefutations of polynomial size. We prove that the binary encoding of thePigeonhole Principle requires exponentially-sized SA refutations, whereas thebinary encoding of the Least Number Principle admits logarithmic rank,polynomially-sized SA refutations. We continue by considering a refutationsystem between SA and Lasserre (Sum-of-Squares). In this system, the LeastNumber Principle requires linear rank while the Pigeonhole Principle becomesconstant rank.

Sherali-Adams and the binary encoding of combinatorial principles