On Sparser Random 3SAT Refutation Algorithms and Feasible Interpolation

We formalize a combinatorial principle, called the 3XOR principle, due to Feige, Kim and Ofek [FKO06], 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 Ω(n) clauses. Such small size refutations would improve the current best (with respect to the clause density) efficient refutation algorithm, which works only for Ω(n) many clauses [FO07]. We then study the proof complexity of the above formulas in weak extensions of cutting planes and resolution. Specifically, we show that there are polynomial-size refutations of the 3XOR principle in resolution operating with disjunctions of quadratic equations (with small integer coefficients), denoted R(quad). 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 [RT08]). This reduces the question of the existence of efficient deterministic refutation algorithms for random 3SAT with n variables and Ω(n) clauses to the question of feasible interpolation of R(quad) and to the weak automatizability of R(lin).