Strength and limitations of Sherali-Adams and Nullstellensatz proof systems

Abstract

We compare the strength of the algebraic proof systems Sherali-Adams ($\mathrm{SA}$) and Nullstellensatz ($\mathrm{NS}$) with Frege-style proof systems. Unlike bounded-depth Frege, $\mathrm{SA}$ has polynomial-size proofs of the pigeonhole principle (PHP). A natural question is whether adding PHP to bounded-depth Frege is enough to simulate $\mathrm{SA}$. We show that $\mathrm{SA}$, with unary integer coefficients, lies strictly between tree-like depth-1 $\text{Frege}+\mathrm{PHP}$ and tree-like $\text{Resolution}$. We introduce a levelled version of PHP ($L\mathrm{PHP}$) and we show that $\mathrm{SA}$ with integer coefficients lies strictly between tree-like depth-1 $\text{Frege}+L\mathrm{PHP}$ and $\text{Resolution}$. Analogous results are shown for $\mathrm{NS}$ using the bijective (i.e. onto and functional) pigeonhole principle and a leveled version of it.