Pairs of SAT Assignment in Random Boolean Formulae

TL;DR

This paper explores the geometric structure of solutions in random K-satisfiability problems, revealing a clustering phenomenon in the distribution of SAT assignments related to Hamming distances, supported by probabilistic methods.

Contribution

It proves the existence of a sharp threshold for x-satisfiability and identifies a gap in the Hamming distance landscape, connecting geometric properties to clustering and statistical physics insights.

Findings

Existence of a sharp threshold for x-satisfiability

Identification of a gap in the Hamming distance landscape

Support for the clustering scenario in SAT solutions

Abstract

We investigate geometrical properties of the random K-satisfiability problem using the notion of x-satisfiability: a formula is x-satisfiable if there exist two SAT assignments differing in Nx variables. We show the existence of a sharp threshold for this property as a function of the clause density. For large enough K, we prove that there exists a region of clause density, below the satisfiability threshold, where the landscape of Hamming distances between SAT assignments experiences a gap: pairs of SAT-assignments exist at small x, and around x=1/2, but they donot exist at intermediate values of x. This result is consistent with the clustering scenario which is at the heart of the recent heuristic analysis of satisfiability using statistical physics analysis (the cavity method), and its algorithmic counterpart (the survey propagation algorithm). The method uses elementary probabilistic arguments (first and second moment methods), and might be useful in other problems of computational and physical interest where similar phenomena appear.