Balanced K-SAT and Biased random K-SAT on trees

TL;DR

This paper analyzes variations of the K-SAT problem on trees, revealing that the SAT-UNSAT transition on Bethe lattices closely matches thresholds on random graphs for small K and aligns with the dynamical thresholds for larger K.

Contribution

It provides exact solutions for balanced and biased K-SAT on trees, connecting these results to known thresholds on random graphs and survey propagation predictions.

Findings

SAT-UNSAT transition matches on Bethe lattice and random graphs for K=2

Transition thresholds are close to survey propagation estimates for K=3

Deviations occur for higher K, but results align with dynamical thresholds

Abstract

We study and solve some variations of the random K-satisfiability problem - balanced K-SAT and biased random K-SAT - on a regular tree, using techniques we have developed earlier(arXiv:1110.2065). In both these problems, as well as variations of these that we have looked at, we find that the SAT-UNSAT transition obtained on the Bethe lattice matches the exact threshold for the same model on a random graph for K=2 and is very close to the numerical value obtained for K=3. For higher K it deviates from the numerical estimates of the solvability threshold on random graphs, but is very close to the dynamical 1-RSB threshold as obtained from the first non-trivial fixed point of the survey propagation algorithm.