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, 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 + )d/ ln d we have non-reconstruction while for k = (1 − )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...

• Type: Journal
• Year: 2015
• Where: APPROX
• Authors: Charilaos Efthymiou
• Keywords: Algorithms, APPROX 2015