Finding a planted clique by adaptive probing
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We consider a variant of the planted clique problem where we are allowed unbounded computational time but can only investigate a small part of the graph by adaptive edge queries. We determine (up to logarithmic factors) the number of queries necessary both for detecting the presence of a planted clique and for finding the planted clique. Specifically, let G ∼ G(n,1/2,k) be a random graph on n vertices with a planted clique of size k. We show that no algorithm that makes at most q = o(n^2 / k^2 + n) adaptive queries to the adjacency matrix of G is likely to find the planted clique. On the other hand, when k ≥ (2+ϵ) _2 n there exists a simple algorithm (with unbounded computational power) that finds the planted clique with high probability by making q = O( (n^2 / k^2) ^2 n + n n) adaptive queries. For detection, the additive n term is not necessary: the number of queries needed to detect the presence of a planted clique is n^2 / k^2 (up to logarithmic factors).
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