Random CNFs are Hard for Cutting Planes

TL;DR

This paper proves that for certain random k-SAT instances, the Cutting Planes proof system requires exponential size to refute unsatisfiable formulas, highlighting the inherent complexity of these problems.

Contribution

It establishes exponential lower bounds on the size of Cutting Planes refutations for random k-SAT with logarithmic k, advancing understanding of proof complexity.

Findings

Cutting Planes proofs need exponential size for certain random k-SAT instances.

The result applies when k is logarithmic in the number of variables.

It links average-case hardness of random k-SAT to proof complexity theory.

Abstract

The random k-SAT model is the most important and well-studied distribution over k-SAT instances. It is closely connected to statistical physics; it is used as a testbench for satisfiability algorithms, and average-case hardness over this distribution has also been linked to hardness of approximation via Feige's hypothesis. We prove that any Cutting Planes refutation for random k-SAT requires exponential size, for k that is logarithmic in the number of variables, in the (interesting) regime where the number of clauses guarantees that the formula is unsatisfiable with high probability.