Random<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>K</mml:mi></mml:math>-satisfiability problem: From an analytic solution to an efficient algorithm

TLDR

The cavity method introduces a survey‑based order parameter that can be computed per sample and serves as a foundation for designing new algorithms for hard combinatorial optimization problems. The study investigates the satisfiability of randomly chosen K‑clause Boolean formulas. The authors employ the zero‑temperature cavity method to derive the phase diagram for the K=3 case. They uncover an intermediate satisfiable phase characterized by many metastable states that slow down search algorithms, and demonstrate a highly efficient algorithm for 3‑SAT.

Abstract

We study the problem of satisfiability of randomly chosen clauses, each with K Boolean variables. Using the cavity method at zero temperature, we find the phase diagram for the K=3 case. We show the existence of an intermediate phase in the satisfiable region, where the proliferation of metastable states is at the origin of the slowdown of search algorithms. The fundamental order parameter introduced in the cavity method, which consists of surveys of local magnetic fields in the various possible states of the system, can be computed for one given sample. These surveys can be used to invent new types of algorithms for solving hard combinatorial optimizations problems. One such algorithm is shown here for the 3-sat problem, with very good performances.