We consider the Sherali-Adams (SA ) refutation system together with the unusual binary encoding of certain combinatorial principles. For the unary encoding of the Pigeonhole Principle and the Least Number Principle, it is known that linear rank is required for refutations in SA, although both admit refutations of polynomial size. We prove that the binary encoding of the Pigeonhole Principle requires exponentially-sized SA refutations, whereas the binary encoding of the Least Number Principle admits logarithmic rank, polynomially-sized SA refutations. We continue by considering a refutation system between SA and Lasserre (Sum-of-Squares). In this system, the unary encoding of the Least Number Principle requires linear rank while the unary encoding of the Pigeonhole Principle becomes constant rank.
CITATION STYLE
Dantchev, S., Ghani, A., & Martin, B. (2020). Sherali-Adams and the Binary Encoding of Combinatorial Principles. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 12118 LNCS, pp. 336–347). Springer Science and Business Media Deutschland GmbH. https://doi.org/10.1007/978-3-030-61792-9_27
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