Balanced K-satisfiability and biased random K-satisfiability on trees

We study and solve some variations of the random K-satisfiability (K-SAT) problem-balanced K-SAT and biased random K-SAT-on a regular tree, using techniques we have developed earlier. In both these problems as well as variations of these that we have looked at, we find that the transition from the satisfiable to the unsatisfiable regime 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 one-step-replica-symmetry-breaking threshold as obtained from the first nontrivial fixed point of the survey propagation algorithm.