Percolation of satisfiability in finite dimensions

TL;DR

This paper explores how percolation theory applies to finite-dimensional Boolean formulas, revealing no satisfiability transition but a logical connectivity transition, with implications for optimization algorithms and disordered materials.

Contribution

It demonstrates the absence of a satisfiability transition in finite dimensions and analyzes the impact of rare regions and boundary effects on computational complexity.

Findings

No satisfiability transition in finite dimensions

Rare regions cause divergent optimization times

Unique thermodynamic ground state in 2D maximum-satisfiability

Abstract

The satisfiability and optimization of finite-dimensional Boolean formulas are studied using percolation theory, rare region arguments, and boundary effects. In contrast with mean-field results, there is no satisfiability transition, though there is a logical connectivity transition. In part of the disconnected phase, rare regions lead to a divergent running time for optimization algorithms. The thermodynamic ground state for the NP-hard two-dimensional maximum-satisfiability problem is typically unique. These results have implications for the computational study of disordered materials.