In this short note, we revisit two hardness measures for resolution proofs: width and asymmetric width. It is known that for every unsatisfiable CNF F,

width(F \derives \Box) \le awidth(F \derives \Box) + max{ awidth(F \derives \Box), width(F)}.

We give a simple direct proof of the upper bound, ... more >>>

Given a MAX-2-SAT instance, we define a local maximum to be an assignment such that changing any single variable reduces the number of satisfied clauses. We consider the question of the number of local maxima hat an instance of MAX-2-SAT can have. We give upper bounds in both the sparse ... more >>>

Mahaney's Theorem states that, assuming P $\neq$ NP, no NP-hard set can have a polynomially bounded number of yes-instances at each input length. We give an exposition of a very simple unpublished proof of Manindra Agrawal whose ideas appear in Agrawal-Arvind ("Geometric sets of low information content," Theoret. Comp. Sci., ... more >>>