Narrow proofs may be maximally long

Albert Atserias, Massimo Lauria, Jakob Nordström

Research output: Contribution to journalArticlepeer-review

21 Citations (SciVal)


We prove that there are 3-conjunctive normal form 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 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.

Original languageEnglish
Article number19
JournalACM Transactions on Computational Logic
Issue number3
Publication statusPublished - 2016 Feb
Externally publishedYes

Subject classification (UKÄ)

  • Computational Mathematics


  • Degree
  • PCR
  • Polynomial calculus
  • Polynomial calculus resolution
  • Proof complexity
  • Resolution
  • SAR
  • Sherali-Adams
  • Width


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