Solving satisfiability problems by fluctuations: The dynamics of stochastic local search algorithms

TL;DR

This paper analyzes the dynamics of stochastic local search algorithms on random satisfiability problems, identifying two regimes based on constraint density and describing how solution times vary from linear to exponential.

Contribution

It provides numerical and approximate analytical descriptions of the algorithm dynamics, revealing the regimes and fluctuation behaviors in satisfiability problems.

Findings

Low constraintness leads to linear-time solutions.

High constraintness results in exponential solution times.

Solution fluctuations are rare but crucial for finding solutions in high constraint regimes.

Abstract

Stochastic local search algorithms are frequently used to numerically solve hard combinatorial optimization or decision problems. We give numerical and approximate analytical descriptions of the dynamics of such algorithms applied to random satisfiability problems. We find two different dynamical regimes, depending on the number of constraints per variable: For low constraintness, the problems are solved efficiently, i.e. in linear time. For higher constraintness, the solution times become exponential. We observe that the dynamical behavior is characterized by a fast equilibration and fluctuations around this equilibrium. If the algorithm runs long enough, an exponentially rare fluctuation towards a solution appears.