Sparser Random 3SAT Refutation Algorithms and the Interpolation Problem

TL;DR

This paper introduces a new combinatorial principle called the 3XOR principle, demonstrating polynomial-size refutations in a weak proof system, and links the complexity of refuting random 3SAT to the interpolation problem and automatizability of related proof systems.

Contribution

It formalizes the 3XOR principle, provides polynomial-size refutations in R(quad), and connects refutation complexity to the interpolation problem and automatizability of R(lin).

Findings

Polynomial-size refutations of the 3XOR principle in R(quad).

Weak automatizability of R(quad) iff R(lin) is weakly automatizable.

Refuting random 3SAT reduces to the interpolation problem of R(quad).

Abstract

We formalize a combinatorial principle, called the 3XOR principle, due to Feige, Kim and Ofek (2006), as a family of unsatisfiable propositional formulas for which refutations of small size in any propositional proof system that possesses the feasible interpolation property imply an efficient deterministic refutation algorithm for random 3SAT with n variables and \Omega(n^{1.4}) clauses. Such small size refutations would improve the state-of-the-art (with respect to the clause density) efficient refutation algorithm, which works only for \Omega(n^{1.5}) many clauses (Feige and Ofek (2007)). We demonstrate polynomial-size refutations of the 3XOR principle in resolution operating with disjunctions of quadratic equations with small integer coefficients, denoted R(quad); this is a weak extension of cutting planes with small coefficients. We show that R(quad) is weakly automatizable iff R(lin) is weakly automatizable, where R(lin) is similar to R(quad) but with linear instead of quadratic equations (introduced in Raz and Tzameret (2008)). This reduces the problem of refuting random 3CNF with n variables and \Omega(n^{1.4}) clauses to the interpolation problem of R(quad) and to the weak automatizability of R(lin).