# Faster Random $k$-CNF Satisfiability

## TL;DR

This paper introduces a faster algorithm for solving random $k$-SAT problems by focusing search efforts on promising assignments, significantly improving runtime over previous methods.

## Contribution

The paper presents a simple yet effective modification to existing algorithms, reducing the runtime for random $k$-SAT by counting satisfied clauses and focusing on promising assignments.

## Key findings

- Achieves a runtime of $O(2^{n (1- ext{Omega}(rac{ ext{lg}^2 k)}{k})})$ for random $k$-SAT.
- Improves upon previous algorithms with a factor of $2^{n ext{Omega}( ext{lg}^2 k)/k}$.
- Demonstrates that assignments with many satisfied clauses are closer to solutions.

## Abstract

We describe an algorithm to solve the problem of Boolean CNF-Satisfiability when the input formula is chosen randomly. We build upon the algorithms of Sch{\"{o}}ning 1999 and Dantsin et al.~in 2002. The Sch{\"{o}}ning algorithm works by trying many possible random assignments, and for each one searching systematically in the neighborhood of that assignment for a satisfying solution. Previous algorithms for this problem run in time $O(2^{n (1- \Omega(1)/k)})$. Our improvement is simple: we count how many clauses are satisfied by each randomly sampled assignment, and only search in the neighborhoods of assignments with abnormally many satisfied clauses. We show that assignments like these are significantly more likely to be near a satisfying assignment. This improvement saves a factor of $2^{n \Omega(\lg^2 k)/k}$, resulting in an overall runtime of $O(2^{n (1- \Omega(\lg^2 k)/k)})$ for random $k$-SAT.