Bounded-depth Frege lower bounds for weaker pigeonhole principles

We prove a quasi-polynomial lower bound on the size of bounded-depth Frege proofs of the pigeon-hole principle PHPm n where m = (1 + 1=polylog n)n. This lower bound qualitatively matches the knownquasi-polynomial-size bounded-depth Frege proofs for these principles. Our technique, which uses a switching lemma argument like other lower bounds for bounded-depth Frege proofs, is novel in that thetautology to which this switching lemma is applied remains random throughout the argument

Rank bounds and integrality gaps for cutting planes procedures

We present a new method for proving rank lower bounds for the cutting planes procedures of Gomory and Chvátal (GC) and Lovász and Schrijver (LS), when viewed as proof systems for unsatisfiability. We apply this method to obtain the following new results: First, we prove near-optimal rank bounds for GC and LS proofs for several prominent unsatisfiable CNF examples, including random kCNF formulas and the Tseitin graph formulas. It follows from these lower bounds that a linear number of rounds of GC or LS procedures when applied to the sta^ndard MAXSAT linear relaxation does not reduce the integrality gap. Second, we give unsatisfiable examples that have constant rank GC and LS proofs but that require linear rank Resolution proofs. Third, we give examples where the GC rank is O(logn) but the LS rank is linear. Finally, we address the question of siz