Random k-SAT and the Power of Two Choices

TL;DR

This paper extends Achlioptas processes to random k-SAT, proving the existence of rules that can shift the satisfiability threshold for k >= 3, and explores implications for the hardness of random k-SAT decision problems.

Contribution

It introduces a semi-random Achlioptas-process model for k-SAT, demonstrating how specific rules can alter the satisfiability threshold for k ≥ 3, a novel extension in probabilistic combinatorics.

Findings

Existence of rules that shift the k-SAT satisfiability threshold for k ≥ 3

First proof of threshold shift in random k-SAT via Achlioptas processes

Proposed a gap decision problem to study the hardness of random k-SAT

Abstract

We study an Achlioptas-process version of the random k-SAT process: a bounded number of k-clauses are drawn uniformly at random at each step, and exactly one added to the growing formula according to a particular rule. We prove the existence of a rule that shifts the satisfiability threshold. This extends a well-studied area of probabilistic combinatorics (Achlioptas processes) to random CSP's. In particular, while a rule to delay the 2-SAT threshold was known previously, this is the first proof of a rule to shift the threshold of k-SAT for k >= 3. We then propose a gap decision problem based upon this semi-random model. The aim of the problem is to investigate the hardness of the random k-SAT decision problem, as opposed to the problem of finding an assignment or certificate of unsatisfiability. Finally, we discuss connections to the study of Achlioptas random graph processes.