The Threshold for Random k-SAT is 2^k ln2 - O(k)

TL;DR

This paper establishes an explicit lower bound for the satisfiability threshold in random k-SAT problems, improving previous bounds for k>3 and providing precise estimates for specific k values.

Contribution

It introduces a new technique that yields an explicit lower bound for the random k-SAT threshold, surpassing all prior bounds for k>3.

Findings

Lower bound for k=10 is 704.94

Threshold for unsatisfiability is at r > 2^k ln 2

New bounds improve upon previous results for k>3

Abstract

Let F be a random k-SAT formula on n variables, formed by selecting uniformly and independently m = rn out of all possible k-clauses. It is well-known that if r>2^k ln 2, then the formula F is unsatisfiable with probability that tends to 1 as n tends to infinity. We prove that there exists a sequence t_k = O(k) such that if r < 2^k ln 2 - t_k, then the formula F is satisfiable with probability that tends to 1 as n tends to infinity. Our technique yields an explicit lower bound for the random k-SAT threshold for every k. For k>3 this improves upon all previously known lower bounds. For example, when k=10 our lower bound is 704.94 while the upper bound is 708.94.