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

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.

💡 Research Summary

This paper investigates two variants of the random K‑satisfiability (K‑SAT) problem—balanced K‑SAT and biased random K‑SAT—on a regular tree (Bethe lattice) using the recursive message‑passing framework introduced in the authors’ earlier work (arXiv:1110.2065). The authors first define the models. In balanced K‑SAT each clause contains exactly K/2 positive literals and K/2 negative literals (for even K), thereby enforcing a strict symmetry between the two polarities. In biased random K‑SAT a parameter p∈