The asymptotics of the clustering transition for random constraint satisfaction problems
share
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 α = 2^k-1/k (ln k + lnln k + γ_ d + o(1)), c= q (ln q + lnln q + γ_ d+ o(1)), 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.
READ FULL TEXT