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Register a new userThe large deviations of the whitening process in random constraint satisfaction problems
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Random constraint satisfaction problems undergo several phase transitions as the ratio between the number of constraints and the number of variables is varied. When this ratio exceeds the satisfiability threshold no more solutions exist; the satisfiable phase, for less constrained problems, is itself divided in an unclustered regime and a clustered one. In the latter solutions are grouped in clusters of nearby solutions separated in configuration space from solutions of other clusters. In addition the rigidity transition signals the appearance of so-called frozen variables in typical solutions: beyond this threshold most solutions belong to clusters with an extensive number of variables taking the same values in all solutions of the cluster. In this paper we refine the description of this phenomenon by estimating the location of the freezing transition, corresponding to the disappearance of all unfrozen solutions (not only typical ones). We also unveil phase transitions for the existence and uniqueness of locked solutions, in which all variables are frozen. From a technical point of view we characterize atypical solutions with a number of frozen variables different from the typical value via a large deviation study of the dynamics of a stripping process (whitening) that unveils the frozen variables of a solution, building upon recent works on atypical trajectories of the bootstrap percolation dynamics. Our results also bear some relevance from an algorithmic perspective, previous numerical studies having shown that heuristic algorithms of various kinds usually output unfrozen solutions.
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We study the phase diagram and the algorithmic hardness of the random `locked constraint satisfaction problems, and compare them to the commonly studied non-locked problems like satisfiability of boolean formulas or graph coloring. The special property of the locked problems is that clusters of solutions are isolated points. This simplifies significantly the determination of the phase diagram, which makes the locked problems particularly appealing from the mathematical point of view. On the other hand we show empirically that the clustered phase of these problems is extremely hard from the algorithmic point of view: the best known algorithms all fail to find solutions. Our results suggest that the easy/hard transition (for currently known algorithms) in the locked problems coincides with the clustering transition. These should thus be regarded as new benchmarks of really hard 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.
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 $alpha$ 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 $alpha = frac{2^{k-1}}{k} (ln k + ln ln k + gamma_{rm d} + o(1))$, $c= q (ln q + ln ln q + gamma_{rm d}+ o(1))$, where $gamma_{rm d}$ is the same constant for both models. We characterize $gamma_{rm d}$ via a functional equation, solve the latter numerically to estimate $gamma_{rm d} approx 0.871$, and obtain an analytic lowerbound $gamma_{rm d} ge 1 + ln (2 (sqrt{2}-1)) approx 0.812$. Our analysis unveils a subtle interplay of the clustering transition with the rigidity (naive reconstruction) threshold that occurs on the same asymptotic scale at $gamma_{rm r}=1$.
We review the understanding of the random constraint satisfaction problems, focusing on the q-coloring of large random graphs, that has been achieved using the cavity method of the physicists. We also discuss the properties of the phase diagram in temperature, the connections with the glass transition phenomenology in physics, and the related algorithmic issues.
We study in this paper the structure of solutions in the random hypergraph coloring problem and the phase transitions they undergo when the density of constraints is varied. Hypergraph coloring is a constraint satisfaction problem where each constraint includes $K$ variables that must be assigned one out of $q$ colors in such a way that there are no monochromatic constraints, i.e. there are at least two distinct colors in the set of variables belonging to every constraint. This problem generalizes naturally coloring of random graphs ($K=2$) and bicoloring of random hypergraphs ($q=2$), both of which were extensively studied in past works. The study of random hypergraph coloring gives us access to a case where both the size $q$ of the domain of the variables and the arity $K$ of the constraints can be varied at will. Our work provides explicit values and predictions for a number of phase transitions that were discovered in other constraint satisfaction problems but never evaluated before in hypergraph coloring. Among other cases we revisit the hypergraph bicoloring problem ($q=2$) where we find that for $K=3$ and $K=4$ the colorability threshold is not given by the one-step-replica-symmetry-breaking analysis as the latter is unstable towards more levels of replica symmetry breaking. We also unveil and discuss the coexistence of two different 1RSB solutions in the case of $q=2$, $K ge 4$. Finally we present asymptotic expansions for the density of constraints at which various phase transitions occur, in the limit where $q$ and/or $K$ diverge.
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