Finding a planted clique by adaptive probing

Abstract

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 $G\sim G(n,1/2,k)$ 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 $k\ge (2+\epsilon ){\log }_{2}n$ there exists a simple algorithm (with unbounded computational power) that finds the planted clique with high probability by making $q=O((n^2/k^2){\log }^{2}n+n\log n)$ 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).

14 pages, 1 figure