On the freezing of variables in random constraint satisfaction problems

The set of solutions of random constraint satisfaction problems (zero energy groundstates of mean-field diluted spin glasses) undergoes several structural phase transitions as the amount of constraints is increased. This set first breaks down into a large number of well separated clusters. At the freezing transition, which is in general distinct from the clustering one, some variables (spins) take the same value in all solutions of a given cluster. In this paper we study the critical behavior around the freezing transition, which appears in the unfrozen phase as the divergence of the sizes of the rearrangements induced in response to the modification of a variable. The formalism is developed on generic constraint satisfaction problems and applied in particular to the random satisfiability of boolean formulas and to the coloring of random graphs. The computation is first performed in random tree ensembles, for which we underline a connection with percolation models and with the reconstruction problem of information theory. The validity of these results for the original random ensembles is then discussed in the framework of the cavity method.Comment: 32 pages, 7 figure

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

The asymptotics of the clustering transition for random constraint satisfaction problems

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 $\gamma_{\rm r}=1$.Comment: 35 pages, 8 figure

Phylogenetic CSPs are Approximation Resistant

We study the approximability of a broad class of computational problems -- originally motivated in evolutionary biology and phylogenetic reconstruction -- concerning the aggregation of potentially inconsistent (local) information about $n$ items of interest, and we present optimal hardness of approximation results under the Unique Games Conjecture. The class of problems studied here can be described as Constraint Satisfaction Problems (CSPs) over infinite domains, where instead of values $\{0,1\}$ or a fixed-size domain, the variables can be mapped to any of the $n$ leaves of a phylogenetic tree. The topology of the tree then determines whether a given constraint on the variables is satisfied or not, and the resulting CSPs are called Phylogenetic CSPs. Prominent examples of Phylogenetic CSPs with a long history and applications in various disciplines include: Triplet Reconstruction, Quartet Reconstruction, Subtree Aggregation (Forbidden or Desired). For example, in Triplet Reconstruction, we are given $m$ triplets of the form $ij|k$ (indicating that ``items $i,j$ are more similar to each other than to $k$'') and we want to construct a hierarchical clustering on the $n$ items, that respects the constraints as much as possible. Despite more than four decades of research, the basic question of maximizing the number of satisfied constraints is not well-understood. The current best approximation is achieved by outputting a random tree (for triplets, this achieves a 1/3 approximation). Our main result is that every Phylogenetic CSP is approximation resistant, i.e., there is no polynomial-time algorithm that does asymptotically better than a (biased) random assignment. This is a generalization of the results in Guruswami, Hastad, Manokaran, Raghavendra, and Charikar (2011), who showed that ordering CSPs are approximation resistant (e.g., Max Acyclic Subgraph, Betweenness).Comment: 45 pages, 11 figures, Abstract shortened for arxi

Reconstruction of Random Colourings

Reconstruction problems have been studied in a number of contexts including biology, information theory and and statistical physics. We consider the reconstruction problem for random $k$-colourings on the $\Delta$-ary tree for large $k$. Bhatnagar et. al. showed non-reconstruction when $\Delta \leq \frac12 k\log k - o(k\log k)$ and reconstruction when $\Delta \geq k\log k + o(k\log k)$. We tighten this result and show non-reconstruction when $\Delta \leq k[\log k + \log \log k + 1 - \ln 2 -o(1)]$ and reconstruction when $\Delta \geq k[\log k + \log \log k + 1+o(1)]$.Comment: Added references, updated notatio

Reconstruction Threshold for the Hardcore Model

In this paper we consider the reconstruction problem on the tree for the hardcore model. We determine new bounds for the non-reconstruction regime on the k-regular tree showing non-reconstruction when lambda < (ln 2-o(1))ln^2(k)/(2 lnln(k)) improving the previous best bound of lambda < e-1. This is almost tight as reconstruction is known to hold when lambda> (e+o(1))ln^2(k). We discuss the relationship for finding large independent sets in sparse random graphs and to the mixing time of Markov chains for sampling independent sets on trees.Comment: 14 pages, 2 figure

Transductive-Inductive Cluster Approximation Via Multivariate Chebyshev Inequality

Approximating adequate number of clusters in multidimensional data is an open area of research, given a level of compromise made on the quality of acceptable results. The manuscript addresses the issue by formulating a transductive inductive learning algorithm which uses multivariate Chebyshev inequality. Considering clustering problem in imaging, theoretical proofs for a particular level of compromise are derived to show the convergence of the reconstruction error to a finite value with increasing (a) number of unseen examples and (b) the number of clusters, respectively. Upper bounds for these error rates are also proved. Non-parametric estimates of these error from a random sample of sequences empirically point to a stable number of clusters. Lastly, the generalization of algorithm can be applied to multidimensional data sets from different fields.Comment: 16 pages, 5 figure