The Power of Choice for Random Satisfiability

TL;DR

This paper investigates how on-line choices in semi-random k-SAT formulas can delay or hasten the satisfiability transition, demonstrating that a small number of choices significantly impacts the threshold.

Contribution

It introduces Achlioptas processes for k-SAT, showing that a limited number of on-line clause choices can effectively raise or lower the satisfiability threshold.

Findings

Three choices suffice to raise the threshold for all k >= 3.

Two choices suffice to raise the threshold for 3 <= k <= 25.

Two choices suffice to lower the threshold for all k >= 3.

Abstract

We consider Achlioptas processes for k-SAT formulas. We create a semi-random formula with n variables and m clauses, where each clause is a choice, made on-line, between two or more uniformly random clauses. Our goal is to delay the satisfiability/unsatisfiability transition, keeping the formula satisfiable up to densities m/n beyond the satisfiability threshold alpha_k for random k-SAT. We show that three choices suffice to raise the threshold for any k >= 3, and that two choices suffice for all 3 <= k <= 25. We also show that two choices suffice to lower the threshold for all k >= 3, making the formula unsatisfiable at a density below alpha_k.