Narrow Proofs May Be Maximally Long

@article{Atserias2014NarrowPM,
  title={Narrow Proofs May Be Maximally Long},
  author={Albert Atserias and Massimo Lauria and Jakob Nordstr{\"o}m},
  journal={2014 IEEE 29th Conference on Computational Complexity (CCC)},
  year={2014},
  pages={286-297}
}
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 nΩ(w). This shows that the simple counting argument that any formula refutable in width w must have a proof in size nO(w) is essentially tight. Moreover, our lower bounds can be generalized to polynomial calculus resolution (PCR) and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight as well. Our results do… 

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It is shown that the simple counting argument that any formula refutable in width w must have a proof in size n is essentially tight, and the lower bounds can be generalized to polynomial calculus resolution and Sherali-Adams, implying that the corresponding size upper bounds in terms of degree and rank are tight.

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