Joshua Buresh-Oppenheim

Rank bounds and integrality gaps for cutting planes procedures

We present a new method for proving rank lower bounds for Cutting Planes (CP) and several procedures based on lifting due to Lovasz 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 Cutting Planes and Lovasz-Schrijver 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 CP or LS procedures when applied to relaxations of integer linear programs is not sufficient for reducing the integrality gap. Secondly, we give unsatisfiable examples that have constant rank CP and LS proofs but that require linear rank resolution proofs. Thirdly, we give examples where the CP rank is O(log n) but the LS rank is linear. Finally, we address the question of size versus rank: we show that, for both proof systems, rank does not accurately reflect proof size. Specifically, there are examples with polynomial-size CP/LS proofs, but requiring linear rank.

Homogenization and the polynomial calculus

AbstractIn standard implementations of the Gröbner basis algorithm, the original polynomials are homogenized so that each term in a given polynomial has the same degree. In this paper, we study the effect of homogenization on the proof complexity of refutations of polynomials derived from Boolean formulas in both the Polynomial Calculus (PC) and Nullstellensatz systems. We show that the PC refutations of homogenized formulas give crucial information about the complexity of the original formulas. The minimum PC refutation degree of homogenized formulas is equal to the Nullstellensatz refutation degree of the original formulas, whereas the size of the homogenized PC refutation is equal to the size of the PC refutation for the originals. Using this relationship, we prove nearly linear (Ω(n/log n) vs. O(1)) separations between Nullstellensatz and PC degree, for a family of explicitly constructed contradictory 3CNF formulas. Previously, an Ω(n1/2) separation had been proved for equations that did not correspond to any CNF formulas, and a log n separation for equations derived from kCNF formulas.

Minimum witnesses for unsatisfiable 2CNFs

We consider the problem of finding the smallest proof of unsatisfiability of a 2CNF formula. In particular, we look at Resolution refutations and at minimum unsatisfiable subsets of the clauses of the CNF. We give a characterization of minimum tree-like Resolution refutations that explains why, to find them, it is not sufficient to find shortest paths in the implication graph of the CNF. The characterization allows us to develop an efficient algorithm for finding a smallest tree-like refutation and to show that the size of such a refutation is a good approximation to the size of the smallest general refutation. We also give a polynomial time dynamic programming algorithm for finding a smallest unsatisfiable subset of the clauses of a 2CNF.

Minimum 2CNF resolution refutations in polynomial time

We present an algorithm for finding a smallest Resolution refutation of any 2CNF in polynomial time.

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 pigeonhole principle

Uniform Hardness Amplification in NP via Monotone Codes

PHP^{m}_n

Relativized NP search problems and propositional proof systems

where

The complexity of resolution refinements

m= (1+1/{\polylog n})n

Making hard problems harder

. This lower bound qualitatively matches the known quasi-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 the tautology to which this switching lemma is applied remains random throughout the argument.