Finding a large hidden clique in a random graph

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Abstract

We consider the following probabilistic model of a graph on n labeled vertices. First choose a random graph G(n, 1/2), and then choose randomly a subset Q of vertices of size k and force it to be a clique by joining every pair of vertices of Q by an edge. The problem is to give a polynomial time algorithm for finding this hidden clique almost surely for various values of k. This question was posed independently, in various variants, by Jerrum and by Kučera. In this paper we present an efficient algorithm for all k > cn0.5, for any fixed c > 0, thus improving the trivial case k > cn0.5(log n)0.5. The algorithm is based on the spectral properties of the graph.

Original languageEnglish
Pages (from-to)457-466
Number of pages10
JournalRandom Structures and Algorithms
Volume13
Issue number3-4
DOIs
StatePublished - 1998

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