The Phase Diagram of 1-in-3 Satisfiability Problem

TL;DR

This paper investigates the phase diagram of the 1-in-3 satisfiability problem, analyzing its computational complexity and structural properties across different parameter regimes using rigorous methods and the cavity approach.

Contribution

It provides a detailed analysis of the phase transitions and complexity regions of the 1-in-3 satisfiability problem, combining rigorous results and cavity method insights.

Findings

Polynomial-time solvable region identified

Hardness region characterized by replica-symmetry-breaking analysis

Predictions for satisfiability transition point discussed

Abstract

We study the typical case properties of the 1-in-3 satisfiability problem, the boolean satisfaction problem where a clause is satisfied by exactly one literal, in an enlarged random ensemble parametrized by average connectivity and probability of negation of a variable in a clause. Random 1-in-3 Satisfiability and Exact 3-Cover are special cases of this ensemble. We interpolate between these cases from a region where satisfiability can be typically decided for all connectivities in polynomial time to a region where deciding satisfiability is hard, in some interval of connectivities. We derive several rigorous results in the first region, and develop the one-step--replica-symmetry-breaking cavity analysis in the second one. We discuss the prediction for the transition between the almost surely satisfiable and the almost surely unsatisfiable phase, and other structural properties of the phase diagram, in light of cavity method results.