The asymptotics of the clustering transition for random constraint satisfaction problems

Abstract

Random Constraint Satisfaction Problems exhibit several phase transitions when their density of constraints is varied. One of these threshold phenomena, known as the clustering or dynamic transition, corresponds to a transition for an information theoretic problem called tree reconstruction. In this article we study this threshold for two CSPs, namely the bicoloring of k-uniform hypergraphs with a density α of constraints, and the q-coloring of random graphs with average degree c. We show that in the large k, q limit the clustering transition occurs for, where γ d is the same constant for both models. We characterize γ d via a functional equation, solve the latter numerically to estimate γ d ≈ 0.871, and obtain an analytic lowerbound γ d ≥ 1 + ln(2( √ 2 − 1)) ≈ 0.812. Our analysis unveils a subtle interplay of the clustering transition with the rigidity (naive reconstruction) threshold that occurs on the same asymptotic scale at γ r = 1. 4.3 Reweighted probability distributions 19 4.4 Numerical resolution 22 4.5 The fixed point equation and the determination of γ d 25 4.6 An analytic lowerbound on γ d 28 5 Conclusion 29

Biased measures for random Constraint Satisfaction Problems: larger interaction range and asymptotic expansion

“…9), an observation that we also made for the other values of b we investigated. We have given analytical arguments in [44] that indeed the plateau value of m n is exactly equal to 1 at γ d for the uniform measure, our numerical results suggest that this remains true when (b, ǫ) = (1, 0), even if we do not have analytical support for this assumption in the general case.…”

“…This Section being rather long and technical we give here, for the convenience of the reader, the main ideas and explain the organization of the forecoming computation, which is the generalization of the one we presented in [44] for the uniform measure. We will focus on the particular form of the function ψ(p) defined in (6), with the two parameters b and ǫ, and start in Sec.…”

“…To handle the difficulty that arises in the intermediate regime γ d < γ < γ r we will adapt the approach we developed in [44] for the uniform measure, introducing a reweighted probability distribution µ n (η) that gives less importance to the typical quasi-hard fields which do not contribute to C n . We define it as…”

Biased measures for random constraint satisfaction problems: larger interaction range and asymptotic expansion

“…In this paper we build upon our previous works [39,44] and generalize them to obtain two main new results. We first introduce a more generic way of weighting the different solutions of an instance of the hypergraph bicoloring problem, that extends the one presented in [39] and incorporates interactions between variables belonging to different hyperedges, and shows that for finite k it allows a further increase of the dynamic threshold α d .…”

“…As a matter of fact the large k behavior of α d is a rather involved asymptotic expansion, even for the uniform measure, and until recently only relatively loose bounds on the asymptotic behavior of α d were known [41][42][43]. We considered this specific problem in [44] and found that the clustering threshold occurs on the scale α ∼ 2 k−1 ( ln k + ln ln k + γ)/k with γ constant, and more precisely that for the uniform measure γ d,u ≈ 0.871, which falls into the range allowed by the previous bounds [41][42][43].…”