We determine the exact freezing threshold, r^f, 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 r^f 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 r^f, 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 r^f the random CSPs can be solved using very simple algorithms, while for densities above r^f one requires more sophisticated techniques in order to deal with frozen clusters.
@cite_39 proves the asymptotic (in @math ) density for the appearance of what they call rigid variables in @math -COL, @math -NAE-SAT and hypergraph 2-colouring (and proves that this is an upper bound for @math -SAT) . The definition of rigid is somewhat weaker than frozen, but a simple modification extends their proof to show the same for frozen vertices. So @cite_39 provided the asymptotic, in @math , location of the freezing threhold for those models . @cite_25 provided the exact location of the threshold for @math -COL, when @math is sufficiently large.
@cite_25 @cite_39
2128738155 2161506909 2042703645 2963702702
We look at a model of random graphs suggested by Gilbert: given an integer @math and @math , scatter @math vertices independently and uniformly on a metric space, and then add edges connecting pairs of vertices of distance less than @math apart.We consider the asymptotics when the metric space is the interval [0, 1], and @math is a function of @math , for @math . We prove that every upwards closed property of (ordered) graphs has at least a weak threshold in this model on this metric space. (But we do find a metric space on which some upwards closed properties do not even have weak thresholds in this model.) We also prove that every upwards closed property with a threshold much above connectivity's threshold has a strong threshold. (But we also find a sequence of upwards closed properties with lower thresholds that are strictly weak.) In this article we propose new methods for computing the asymptotic value for the logarithm of the partition function (free energy) for certain statistical physics models on certain type of finite graphs, as the size of the underlying graph goes to infinity. The two models considered are the hard-core (independent set) model when the activity parameter λ is small, and also the Potts (q-coloring) model. We only consider the graphs with large girth. In particular, we prove that asymptotically the logarithm of the number of independent sets of any r-regular graph with large girth when rescaled is approximately constant if r ≤ 5. For example, we show that every 4-regular n-node graph with large girth has approximately (1.494…)n-many independent sets, for large n. Further, we prove that for every r-regular graph with r ≥ 2, with n nodes and large girth, the number of proper q ≥ r + 1 colorings is approximately @math n, for large n. We also show that these results hold for random regular graphs with high probability (w.h.p.) as well. As a byproduct of our method we obtain simple algorithms for the problem of computing approximately the logarithm of the number of independent sets and proper colorings, in low degree graphs with large girth. These algorithms are deterministic and use certain correlation decay properties for the corresponding Gibbs measures, and its implications to uniqueness of the Gibbs measures on the infinite trees, as well as some simple cavity trick which is well known in the physics and the Markov chain sampling literature.© 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008 Preliminary version of this paper appeared in Proceedings of the Seventeenth ACM-SIAM Symposium on Discrete Algorithms 2006. ArXive version. Abstract We show that if a collection of hypergraphs (1) is uniform (every edge contains exactly k vertices, for some fixed k), (2) has minimum degree asymptotic to the maximum degree, and (3) has maximum codegree (the number of edges containing a pair of vertices) asymptotically negligible compared with the maximum degree, then the chromatic index is asymptotic to the maximum degree. This means that the edges can be partitioned into packings (or matchings), almost all of which are almost perfect. We also show that the edges can be partitioned into coverings, almost all of which are almost perfect. The result strengthens and generalizes a result due to Frankl and Rodl concerning the existence of a single almost perfect packing or covering under similar circumstances. In particular, it shows that the chromatic index of a Steiner triple-system on n points is asymptotic to n 2 , resolving a long-standing conjecture. We prove tight upper bounds on the logarithmic derivative of the independence and matching polynomials of d-regular graphs. For independent sets, this theorem is a strengthening of Kahn's result that a disjoint union of copies of Kd;d maximizes the number of independent sets of a bipartite d-regular graph, Galvin and Tet ali's result that the independence polynomial is maximized by the same, and Zhao's extension of both results to all d-regular graphs. For matchings, this shows that the matching polynomial and the total number of matchings of a d-regular graph are maximized by a union of copies of Kd;d. Using this we prove the asymptotic upper matching conjecture of Friedland, Krop, Lundow, and Markstrom. In probabilistic language, our main theorems state that for all d-regular graphs and all �, the occupancy fraction of the hard-core model and the edge occupancy fraction of the monomer-dimer model with fugacity � are maximized by Kd;d. Our method involves constrained optimization problems over distributions of random variables and applies to all d-regular graphs directly, without a reduction to the bipartite case. Using a variant of the method we prove a lower bound on the occupancy fraction of the hard-core model on any d-regular, vertex-transitive, bipartite graph: the occupancy fraction of such a graph is strictly greater than the occupancy fraction of the unique translationinvariant hard-core measure on the infinite d-regular tree