Maximally flexible solutions of a random $K$-satisfiability formula

TL;DR

This paper investigates maximally flexible solutions in random K-SAT problems, focusing on solutions with the fewest variables satisfying all constraints, revealing dense regions of the solution space and employing advanced statistical and algorithmic methods.

Contribution

It introduces a novel focus on atypical, maximally flexible solutions in K-SAT, estimating their maximum null variable fraction using the replica-symmetric cavity method and constructing them via message-passing algorithms.

Findings

Maximally flexible solutions contain a high fraction of null variables.

The maximum null variable fraction can be estimated using the replica-symmetric cavity method.

Message-passing algorithms can construct such solutions for specific instances.

Abstract

Random $K$-satisfiability ($K$-SAT) is a paradigmatic model system for studying phase transitions in constraint satisfaction problems and for developing empirical algorithms. The statistical properties of the random $K$-SAT solution space have been extensively investigated, but most earlier efforts focused on solutions that are typical. Here we consider maximally flexible solutions which satisfy all the constraints only using the minimum number of variables. Such atypical solutions have high internal entropy because they contain a maximum number of null variables which are completely free to choose their states. Each maximally flexible solution indicates a dense region of the solution space. We estimate the maximum fraction of null variables by the replica-symmetric cavity method, and implement message-passing algorithms to construct maximally flexible solutions for single $K$-SAT instances.