Finite-size scaling in random $K$-satisfiability problems

TL;DR

This paper investigates finite-size scaling in random K-satisfiability problems, revealing how phase transitions can be characterized by the density of unsatisfied clauses and proposing conjectures for the FSS exponent.

Contribution

It introduces a finite-size scaling analysis for phase transitions in random K-satisfiability, supported by numerical simulations and clustering arguments.

Findings

Density of unsatisfied clauses signals phase transition.

Two conjectured values for the FSS exponent.

Numerical simulations support the conjectures for K=2 and 3.

Abstract

We provide a comprehensive view of various phase transitions in random $K$-satisfiability problems solved by stochastic-local-search algorithms. In particular, we focus on the finite-size scaling (FSS) exponent, which is mathematically important and practically useful in analyzing finite systems. Using the FSS theory of nonequilibrium absorbing phase transitions, we show that the density of unsatisfied clauses clearly indicates the transition from the solvable (absorbing) phase to the unsolvable (active) phase as varying the noise parameter and the density of constraints. Based on the solution clustering (percolation-type) argument, we conjecture two possible values of the FSS exponent, which are confirmed reasonably well in numerical simulations for $2\le K \le 3$.