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Reconstruction/Non-reconstruction Thresholds for Colourings of General Galton-Watson Trees

Author Charilaos Efthymiou

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Charilaos Efthymiou

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Charilaos Efthymiou. Reconstruction/Non-reconstruction Thresholds for Colourings of General Galton-Watson Trees. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 40, pp. 756-774, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2015.756

Abstract

The broadcasting models on trees arise in many contexts such as discrete mathematics, biology, information theory, statistical physics and computer science. In this work, we consider the k-colouring model. A basic question here is whether the assignment at the root affects the distribution  of the colourings at the vertices at distance h from the root. This is the so-called reconstruction  problem. For the case where the underlying tree is d -ary it is well known that d/ln(d) is the reconstruction threshold. That is, for k=(1+epsilon)*d/ln(d) we have non-reconstruction while for k=(1-epsilon)*d/ln(d)  we have reconstruction.

Here, we consider the largely unstudied  case where the underlying tree is chosen according to a predefined distribution. In particular, we consider the well-known Galton-Watson trees. The corresponding model arises naturally in many contexts such as
the theory of spin-glasses and its applications on random Constraint Satisfaction Problems (rCSP). The study on rCSP 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).

Subject Classification

Keywords
  • Random Colouring
  • Reconstruction Problem
  • Galton-Watson Tree

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References

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