Critical behaviour of combinatorial search algorithms, and the unitary-propagation universality class

TL;DR

This paper investigates the success probability of search algorithms for random SAT problems, revealing a universal critical behavior near a threshold and deriving an exact scaling function through a novel mapping.

Contribution

It demonstrates the universality of critical behavior for algorithms based on unitary propagation and provides an exact calculation of the scaling function Phi.

Findings

Success probability drops exponentially above threshold

Universal critical exponents related to random graph behavior

Exact form of the scaling function Phi derived

Abstract

The probability P(alpha, N) that search algorithms for random Satisfiability problems successfully find a solution is studied as a function of the ratio alpha of constraints per variable and the number N of variables. P is shown to be finite if alpha lies below an algorithm--dependent threshold alpha\_A, and exponentially small in N above. The critical behaviour is universal for all algorithms based on the widely-used unitary propagation rule: P[ (1 + epsilon) alpha\_A, N] ~ exp[-N^(1/6) Phi(epsilon N^(1/3)) ]. Exponents are related to the critical behaviour of random graphs, and the scaling function Phi is exactly calculated through a mapping onto a diffusion-and-death problem.