On the behaviour of randomK-SAT on trees

Abstract

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 → ∞ 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.

BalancedK-satisfiability and biased randomK-satisfiability on trees

“…For higher K, we have studied the problem numerically on regular random graphs and find that the threshold estimated on the tree is very close to the threshold obtained numerically for K = 3, although deviating more and more as K increases. Interestingly, as before [14], the values obtained by our method are also very close to the value predicted for the clustering transition by dynamical one-step-replicasymmetry-breaking (1-RSB) calculations on a random graph for these models [15,16]. In this paper, we suggest an explanation for this fact by working out a connection between our approach and survey propagation (SP).…”

“…For the problems we looked at, we are interested in the recursions for these quantities deep within the tree as in Ref. [14] so that we can get rid of boundary effects.…”

“…We consider a model where each variable has a fixed degree d + 1 and a variable appears as non-negated with probability p. For p = 1/2, this corresponds to the uniform random K-SAT [14]. This model was also studied in Ref.…”

“…Recently, we have studied the random K-SAT defined on a regular d-ary rooted tree [14]. By fixing the boundary, we could calculate the moments of the number of solutions (averaged over all possible instances of the logical expression) exactly.…”

“…This is the natural analog of our tree computation for random graphs. We make this comparison not only for the random K-SAT studied earlier [14], but also for two variants of the random K-SAT, which we solve on a d-ary rooted tree. We find in all cases that, for K = 2, the threshold obtained via the tree calculations matches the known exact value for a regular random graph.…”