文章基本信息

标题：The Lovász Theta Function for Random Regular Graphs and Community Detection in the Hard Regime
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作者：Jess Banks ; Robert Kleinberg ; Cristopher Moore 等
期刊名称：LIPIcs : Leibniz International Proceedings in Informatics
电子版ISSN：1868-8969
出版年度：2017
卷号：81
页码：28:1-28:22
DOI：10.4230/LIPIcs.APPROX-RANDOM.2017.28
出版社：Schloss Dagstuhl -- Leibniz-Zentrum fuer Informatik
摘要：We derive upper and lower bounds on the degree d for which the Lovasz theta function, or equivalently sum-of-squares proofs with degree two, can refute the existence of a k-coloring in random regular graphs G(n,d). We show that this type of refutation fails well above the k-colorability transition, and in particular everywhere below the Kesten-Stigum threshold. This is consistent with the conjecture that refuting k-colorability, or distinguishing G(n,d) from the planted coloring model, is hard in this region. Our results also apply to the disassortative case of the stochastic block model, adding evidence to the conjecture that there is a regime where community detection is computationally hard even though it is information-theoretically possible. Using orthogonal polynomials, we also provide explicit upper bounds on the theta function for regular graphs of a given girth, which may be of independent interest.
关键词：Lov{\'a