We study the effect of boundary conditions on the relaxation time (i.e., inverse spectral gap) of the Glauber dynamics for the hard-core model on the tree. The hard-core model is defined on the set of independent sets weighted by a parameter $\lambda$, called the activity or fugacity. The Glauber dynamics is the Markov chain that updates a randomly chosen vertex in each step. On the infinite tree with branching factor $b$, the hard-core model can be equivalently defined as a broadcasting process with a parameter $\omega$ which is the positive solution to $\lambda=\omega(1+\omega)^b$, and vertices are occupied with probability $\omega/(1+\omega)$ when their parent is unoccupied. This broadcasting process undergoes a phase transition between the so-called reconstruction and nonreconstruction regions at $\omega_r\approx \ln{b}/b$. Reconstruction has been of considerable interest recently since it appears to be intimately connected to the efficiency of local algorithms on locally tree-like graphs, such as sparse random graphs. In this paper we show that the relaxation time of the Glauber dynamics on regular trees $T_h$ of height $h$ with branching factor $b$ and $n$ vertices undergoes a phase transition around the reconstruction threshold. In particular, we construct a boundary condition for which the relaxation time slows down at the reconstruction threshold. More precisely, for any $\omega \le \ln{b}/b$, for $T_h$ with any boundary condition, the relaxation time is $\Omega(n)$ and $O(n^{1+o_b(1)})$. In contrast, above the reconstruction threshold we show that for every $\delta>0$, for $\omega=(1+\delta)\ln{b}/b$, the relaxation time on $T_h$ with any boundary condition is $O(n^{1+\delta + o_b(1)})$, and we construct a boundary condition where the relaxation time is $\Omega(n^{1+\delta/2 - o_b(1)})$. To prove this lower bound in the reconstruction region we introduce a general technique that transforms a reconstruction algorithm into a set with poor conductance.

We determine the exact freezing threshold, rf, for a family of models of random boolean constraint satisfaction problems, including NAE-SAT and hypergraph 2-colouring, when the constraint size is sufficiently large. If the constraint-density of a random CSP, F, in our family is greater than rf then for almost every solution of F, a linear number of variables are frozen, meaning that their colours cannot be changed by a sequence of alterations in which we change o(n) variables at a time, always switching to another solution. If the constraint-density is less than rf, then almost every solution has o(n) frozen variables. Freezing is a key part of the clustering phenomenon that is hypothesized by non-rigorous techniques from statistical physics. The understanding of clustering has led to the development of advanced heuristics such as Survey Propogation. It has been suggested that the freezing threshold is a precise algorithmic barrier: that for densities below rf the random CSPs can be solved using very simple algorithms, while for densities above rf one requires more sophisticated techniques in order to deal with frozen clusters.

The hard-core model has received much attention in the past couple of decades as a lattice gas model with hard constraints in statistical physics, a multicast model of calls in communication networks, and as a weighted independent set problem in combinatorics, probability and theoretical computer science. In this model, each independent set I in a graph G is weighted proportionally to λ|I|, for a positive real parameter λ. For large λ, computing the partition function (namely, the normalizing constant which makes the weighting a probability distribution on a finite graph) on graphs of maximum degree Δ ≥ 3, is a well known computationally challenging problem. More concretely, let λc(TΔ) denote the critical value for the so-called uniqueness threshold of the hard-core model on the infinite Δ-regular tree; recent breakthrough results of Dror Weitz (2006) and Allan Sly (2010) have identified λc(TΔ) as a threshold where the hardness of estimating the above partition function undergoes a computational transition. We focus on the well-studied particular case of the square lattice Z2, and provide a new lower bound for the uniqueness threshold, in particular taking it well above λc(T4). Our technique refines and builds on the tree of self-avoiding walks approach of Weitz, resulting in a new technical sufficient criterion (of wider applicability) for establishing strong spatial mixing (and hence uniqueness) for the hard-core model. Our new criterion achieves better bounds on strong spatial mixing when the graph has extra structure, improving upon what can be achieved by just using the maximum degree. Applying our technique to Z2we prove that strong spatial mixing holds for all λ <; 2.3882, improving upon the work of Weitz that held for λ <; 27/16 = 1.6875. Our results imply a fully-polynomial deterministic approximation algorithm for estimating the partition function, as well as rapid mixing of the associated Glauber dynamics to sample from the hard-core distribution.

We study the effect of boundary conditions on the relaxation time of the Glauber dynamics for the hardcore lattice gas model on the n-vertex regular b-ary tree of height h. The hard-core model is defined on independent sets weighted by an activity (or fugacity) Λ on trees. Reconstruction studies the effect of a ‘typical’ boundary condition, i.e., fixed assignment to the leaves, on the root. The threshold for when reconstruction occurs (and a typical boundary influences the root in the limit h → ∞) has been of considerable recent interest since it appears to be connected to the efficiency of certain local algorithms on locally tree-like graphs. The reconstruction threshold occurs at ω ≈ ln b/b where Λ = ω(1 + ω)b is a convenient re-parameterization of the model. We prove that for all boundary conditions, the relaxation time τ in the non-reconstruction region is fast, namely τ = O(n1+ob(1)) for any ω ≤ ln b/b. In the reconstruction region, for all boundary conditions, we prove τ = O(n1+δ+ob(1)) for ω = (1 + δ) ln b/b, for every δ > 0. In contrast, we construct a boundary condition, for which the Glauber dynamics slows down in the reconstruction region, namely τ = Ω(n1+δ/2−ob(1)) for ω = (1 + δ) ln b/b, for every δ > 0. The interesting part of our proof is this lower bound result, which uses a general technique that transforms an algorithm to prove reconstruction into a set in the state space of the Glauber dynamics with poor conductance.

Random instances of constraint satisfaction problems (CSPs) appear to be hard for all known algorithms when the number of constraints per variable lies in a certain interval. Contributing to the general understanding of the structure of the solution space of a CSP in the satisfiable regime, we formulate a set of technical conditions on a large family of random CSPs and prove bounds on three most interesting thresholds for the density of such an ensemble: namely, the satisfiability threshold, the threshold for clustering of the solution space, and the threshold for an appropriate reconstruction problem on the CSPs. The bounds become asymptoticlally tight as the number of degrees of freedom in each clause diverges. The families are general enough to include commonly studied problems such as random instances of Not-All-Equal SAT, [Formula: see text]-XOR formulae, hypergraph 2-coloring, and graph [Formula: see text]-coloring. An important new ingredient is a condition involving the Fourier expansion of clauses, which characterizes the class of problems with a similar threshold structure.

We consider the problem of distributing a file in a network of storage nodes whose storage budget is limited but at least equals the size file. We first generate T encoded symbols (from the file) which are then distributed among the nodes. We investigate the optimal allocation of T encoded packets to the storage nodes such that the probability of reconstructing the file by using any r out of n nodes is maximized. Since the optimal allocation of encoded packets is difficult to find in general, we find another objective function which well approximates the original problem and yet is easier to optimize. We find the optimal symmetric allocation for all coding redundancy constraints using the equivalent approximate problem. We also investigate the optimal allocation in random graphs. Finally, we provide simulations to verify the theoretical results.