On the behaviour of random K-SAT on trees

We consider the K-satisfiability problem on a regular d-ary rooted tree. For this model, we demonstrate how we can calculate in closed form, the moments of the total number of solutions as a function of d and K, where the average is over all realizations, for a fixed assignment of the surface variables. We find that different moments pick out different ‘critical’ values of d, below which they diverge as the total number of variables on the tree goes to infinity and above which they decay. We show that K-SAT on the random graph also behaves similarly. We also calculate exactly the fraction of instances that have solutions for all K. On the tree, this quantity decays to 0 (as the number of variables increases) for any d>1. However the recursion relations for this quantity have a non-trivial fixed-point solution which indicates the existence of a different transition in the interior of an infinite rooted tree.

💡 Research Summary

The paper investigates the random K‑SAT problem on a regular d‑ary rooted tree, fixing the values of the variables on the surface (the leaves) and averaging over all possible clause realizations. By defining the total number of satisfying assignments Z_R(L,n) for a tree of depth n with a given boundary condition L, the authors split Z_R into two contributions, F_R(n) and G_R(n), corresponding to the root taking value 0 or 1 respectively. They derive exact recursion relations for these quantities based on the independence of sub‑trees and the random choice of whether each edge is negated (dashed) or not (solid).

The first moment ⟨Z⟩ is obtained directly from the probability that a single clause is satisfied, (2^K‑1)/2^K, leading to ⟨Z⟩ = ((2^K‑1)/2^K·d/2)^{N(n)} where N(n) is the number of variables in a depth‑n tree. This yields a critical branching factor d_c = –log 2 / log(1‑2^{-K}). For d > d_c the average number of solutions grows exponentially with the size of the tree, while for d < d_c it decays, reproducing the annealed result known for random graphs when d is replaced by the clause‑to‑variable ratio α.

The second moment ⟨Z²⟩ requires tracking both ⟨F²⟩ and ⟨F·G⟩. Introducing the ratio r_n = ⟨F·G⟩/⟨F²⟩, the authors obtain a closed recursion r_n =