Narrow proofs may be maximally long

• 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(ω) 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 not extend all the way to Lasserre, however-the formulas we study have Lasserre proofs of constant rank and size polynomial in both n and w.

Subject headings

NATURVETENSKAP  -- Data- och informationsvetenskap (hsv//swe)

NATURAL SCIENCES  -- Computer and Information Sciences (hsv//eng)

Keyword

degree

Lasserre

length

PCR

polynomial calculus

proof complexity

rank

resolution

Sherali-Adams

size

width

Calculations

Computational complexity

Optical resolving power

Polynomials

Publication and Content Type

ref (subject category)

kon (subject category)