Biased landscapes for random Constraint Satisfaction Problems

The typical complexity of Constraint Satisfaction Problems (CSPs) can be investigated by means of random ensembles of instances. The latter exhibit many threshold phenomena besides their satisfiability phase transition, in particular a clustering or dynamic phase transition (related to the tree reconstruction problem) at which their typical solutions shatter into disconnected components. In this paper we study the evolution of this phenomenon under a bias that breaks the uniformity among solutions of one CSP instance, concentrating on the bicoloring of k-uniform random hypergraphs. We show that for small k the clustering transition can be delayed in this way to higher density of constraints, and that this strategy has a positive impact on the performances of Simulated Annealing algorithms. We characterize the modest gain that can be expected in the large k limit from the simple implementation of the biasing idea studied here. This paper contains also a contribution of a more methodological nature, made of a review and extension of the methods to determine numerically the discontinuous dynamic transition threshold.Comment: 32 pages, 16 figure

Rigid colorings of hypergraphs and contiguity

©© 2019 Society for Industrial and Applied Mathematics We consider the problem of q-coloring a k-uniform random hypergraph, where q, k > 3, and determine the rigidity threshold. For edge densities above the rigidity threshold, we show that almost all solutions have a linear number of vertices that are linearly frozen, meaning that they cannot be recolored by a sequence of colorings that each change the color of a sublinear number of vertices. When the edge density is below the threshold, we prove that all but a vanishing proportion of the vertices can be recolored by a sequence of colorings that recolor only one vertex at a time. This change in the geometry of the solution space has been hypothesized to be the cause of the algorithmic barrier faced by naive coloring algorithms. Our calculations verify predictions made by statistical physicists using the nonrigorous cavity method. The traditional model for problems of this type is the random coloring model, where a random hypergraph is chosen and then a random coloring of that hypergraph is selected. However, it is often easier to work with the planted model, where a random coloring is selected first, and then edges are randomly chosen which respect the coloring. As part of our analysis, we show that up to the condensation phase transition, the random coloring model is contiguous with respect to the planted model. This result is of independent interest

One-step replica symmetry breaking of random regular NAE-SAT I

In a broad class of sparse random constraint satisfaction problems(CSP), deep heuristics from statistical physics predict that there is a condensation phase transition before the satisfiability threshold, governed by one-step replica symmetry breaking(1RSB). In fact, in random regular k-NAE-SAT, which is one of such random CSPs, it was verified \cite{ssz16} that its free energy is well-defined and the explicit value follows the 1RSB prediction. However, for any model of sparse random CSP, it has been unknown whether the solution space indeed condensates on O(1) clusters according to the 1RSB prediction. In this paper, we give an affirmative answer to this question for the random regular k-NAE-SAT model. Namely, we prove that with probability bounded away from zero, most of the solutions lie inside a bounded number of solution clusters whose sizes are comparable to the scale of the free energy. Furthermore, we establish that the overlap between two independently drawn solutions concentrates precisely at two values. Our proof is based on a detailed moment analysis of a spin system, which has an infinite spin space that encodes the structure of solution clusters. We believe that our method is applicable to a broad range of random CSPs in the 1RSB universality class.Comment: The previous version is divided into two parts and this submission is Part I of a two-paper serie