Criticality and Universality in the Unit-Propagation Search Rule

TL;DR

This paper investigates the success probability of stochastic greedy algorithms with unit-propagation in solving random SAT problems, revealing universal critical behavior and phase transitions depending on input parameters.

Contribution

It identifies the universal critical behavior and universality classes of success probability in SAT algorithms using unit-propagation, with detailed phase diagrams and scaling laws.

Findings

Two universality classes identified: power law and stretched exponential.

Critical exponent gamma is universal and explicitly calculated.

Scaling functions derived from reaction-diffusion equations match numerical results.

Abstract

The probability Psuccess(alpha, N) that stochastic greedy algorithms successfully solve the random SATisfiability problem is studied as a function of the ratio alpha of constraints per variable and the number N of variables. These algorithms assign variables according to the unit-propagation (UP) rule in presence of constraints involving a unique variable (1-clauses), to some heuristic (H) prescription otherwise. In the infinite N limit, Psuccess vanishes at some critical ratio alpha\_H which depends on the heuristic H. We show that the critical behaviour is determined by the UP rule only. In the case where only constraints with 2 and 3 variables are present, we give the phase diagram and identify two universality classes: the power law class, where Psuccess[alpha\_H (1+epsilon N^{-1/3}), N] ~ A(epsilon)/N^gamma; the stretched exponential class, where Psuccess[alpha\_H (1+epsilon N^{-1/3}), N] ~ exp[-N^{1/6} Phi(epsilon)]. Which class is selected depends on the characteristic parameters of input data. The critical exponent gamma is universal and calculated; the scaling functions A and Phi weakly depend on the heuristic H and are obtained from the solutions of reaction-diffusion equations for 1-clauses. Computation of some non-universal corrections allows us to match numerical results with good precision. The critical behaviour for constraints with >3 variables is given. Our results are interpreted in terms of dynamical graph percolation and we argue that they should apply to more general situations where UP is used.