95 research outputs found
Reconstruction/Non-reconstruction Thresholds for Colourings of General Galton-Watson Trees
The broadcasting models on trees arise in many contexts such as discrete
mathematics, biology statistical physics and cs. In this work, we consider the
colouring model. A basic question here is whether the root's assignment affects
the distribution of the colourings at the vertices at distance h from the root.
This is the so-called "reconstruction problem". For a d-ary tree it is well
known that d/ln (d) is the reconstruction threshold. That is, for
k=(1+eps)d/ln(d) we have non-reconstruction while for k=(1-eps)d/ln(d) we have.
Here, we consider the largely unstudied case where the underlying tree is
chosen according to a predefined distribution. In particular, our focus is on
the well-known Galton-Watson trees. This model arises naturally in many
contexts, e.g. the theory of spin-glasses and its applications on random
Constraint Satisfaction Problems (rCSP). The aforementioned study focuses on
Galton-Watson trees with offspring distribution B(n,d/n), i.e. the binomial
with parameters n and d/n, where d is fixed. Here we consider a broader version
of the problem, as we assume general offspring distribution, which includes
B(n,d/n) as a special case.
Our approach relates the corresponding bounds for (non)reconstruction to
certain concentration properties of the offspring distribution. This allows to
derive reconstruction thresholds for a very wide family of offspring
distributions, which includes B(n,d/n). A very interesting corollary is that
for distributions with expected offspring d, we get reconstruction threshold
d/ln(d) under weaker concentration conditions than what we have in B(n,d/n).
Furthermore, our reconstruction threshold for the random colorings of
Galton-Watson with offspring B(n,d/n), implies the reconstruction threshold for
the random colourings of G(n,d/n)
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 -colourings on the -ary tree for
large . Bhatnagar et. al. showed non-reconstruction when and reconstruction when . We tighten this result and show non-reconstruction when and reconstruction when .Comment: Added references, updated notatio
Uniformly Random Colourings of Sparse Graphs
We analyse uniformly random proper -colourings of sparse graphs with
maximum degree in the regime . This regime
corresponds to the lower side of the shattering threshold for random graph
colouring, a paradigmatic example of the shattering threshold for random
Constraint Satisfaction Problems. We prove a variety of results about the
solution space geometry of colourings of fixed graphs, generalising work of
Achlioptas, Coja-Oghlan, and Molloy on random graphs, and justifying the
performance of stochastic local search algorithms in this regime. Our central
proof relies only on elementary techniques, namely the first-moment method and
a quantitative induction, yet it strengthens list-colouring results due to Vu,
and more recently Davies, Kang, P., and Sereni, and generalises
state-of-the-art bounds from Ramsey theory in the context of sparse graphs. It
further yields an approximately tight lower bound on the number of colourings,
also known as the partition function of the Potts model, with implications for
efficient approximate counting
On a Connectivity Threshold for Colorings of Random Graphs and Hypergraphs
Let Omega_q=Omega_q(H) denote the set of proper [q]-colorings of the hypergraph H. Let Gamma_q be the graph with vertex set Omega_q where two vertices are adjacent iff the corresponding colorings differ in exactly one vertex. We show that if H=H_{n,m;k}, k >= 2, the random k-uniform hypergraph with V=[n] and m=dn/k hyperedges then w.h.p. Gamma_q is connected if d is sufficiently large and q >~ (d/log d)^{1/(k-1)}. This is optimal to the first order in d. Furthermore, with a few more colors, we find that the diameter of Gamma_q is O(n) w.h.p, where the hidden constant depends on d. So, with this choice of d,q, the natural Glauber Dynamics Markov Chain on Omega_q is ergodic w.h.p
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
Local convergence of random graph colorings
Let be a random graph whose average degree is below the
-colorability threshold. If we sample a -coloring of
uniformly at random, what can we say about the correlations between the colors
assigned to vertices that are far apart? According to a prediction from
statistical physics, for average degrees below the so-called {\em condensation
threshold} , the colors assigned to far away vertices are
asymptotically independent [Krzakala et al.: Proc. National Academy of Sciences
2007]. We prove this conjecture for exceeding a certain constant .
More generally, we investigate the joint distribution of the -colorings that
induces locally on the bounded-depth neighborhoods of any fixed number
of vertices. In addition, we point out an implication on the reconstruction
problem
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
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