Narrow Proofs May Be Maximally Long

Abstract

We prove that there are 3-CNF formulas over n variables that can be refuted in resolution in width w but require resolution proofs of size n Ω( w ) . This shows that the simple counting argument that any formula refutable in width w must have a proof in size n O( w ) is essentially tight. Moreover, our lower bound generalizes to polynomial calculus resolution and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. The lower bound does not extend all the way to Lasserre, however, since we show that there the formulas we study have proofs of constant rank and size polynomial in both n and w .