{"@context":"https:\/\/schema.org\/","@type":"ScholarlyArticle","@id":"#article8083","name":"Reconstruction\/Non-reconstruction Thresholds for Colourings of General Galton-Watson Trees","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.\r\n\r\nHere, 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\r\nthe 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.\r\n\r\nOur 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).\r\n \r\nFurthermore, 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).","keywords":["Random Colouring","Reconstruction Problem","Galton-Watson Tree"],"author":{"@type":"Person","name":"Efthymiou, Charilaos","givenName":"Charilaos","familyName":"Efthymiou"},"position":44,"pageStart":756,"pageEnd":774,"dateCreated":"2015-08-13","datePublished":"2015-08-13","isAccessibleForFree":true,"license":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/legalcode","copyrightHolder":{"@type":"Person","name":"Efthymiou, Charilaos","givenName":"Charilaos","familyName":"Efthymiou"},"copyrightYear":"2015","accessMode":"textual","accessModeSufficient":"textual","creativeWorkStatus":"Published","inLanguage":"en-US","sameAs":"https:\/\/doi.org\/10.4230\/LIPIcs.APPROX-RANDOM.2015.756","publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","isPartOf":{"@type":"PublicationVolume","@id":"#volume6243","volumeNumber":40,"name":"Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX\/RANDOM 2015)","dateCreated":"2015-08-13","datePublished":"2015-08-13","editor":[{"@type":"Person","name":"Garg, Naveen","givenName":"Naveen","familyName":"Garg"},{"@type":"Person","name":"Jansen, Klaus","givenName":"Klaus","familyName":"Jansen"},{"@type":"Person","name":"Rao, Anup","givenName":"Anup","familyName":"Rao"},{"@type":"Person","name":"Rolim, Jos\u00e9 D. P.","givenName":"Jos\u00e9 D. P.","familyName":"Rolim"}],"isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#article8083","isPartOf":{"@type":"Periodical","@id":"#series116","name":"Leibniz International Proceedings in Informatics","issn":"1868-8969","isAccessibleForFree":true,"publisher":"Schloss Dagstuhl \u2013 Leibniz-Zentrum f\u00fcr Informatik","hasPart":"#volume6243"}}}