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

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 [14]. In both these problems, as well as variations of these that we have looked at, we find that the SATUNSAT 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.