On semantic cutting planes with very small coefficients

Cutting planes proofs for integer programs can naturally be defined both in a syntactic and in a semantic fashion. Filmus et al. (STACS 2016) proved that semantic cutting planes proofs may be exponentially stronger than syntactic ones, even if they use the semantic rule only once. We show that when semantic cutting planes proofs are restricted to have coefficients bounded by a function growing slowly enough, syntactic cutting planes can simulate them efficiently. Furthermore if we strengthen the restriction to a constant bound, then the simulating syntactic proof even has polynomially small coefficients.