Finding Planted Cliques in Sublinear Time
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We study the planted clique problem in which a clique of size k is planted in an Erdős-Rényi graph of size n and one wants to recover this planted clique. For k=Ω(√(n)), polynomial time algorithms can find the planted clique. The fastest such algorithms run in time linear O(n^2) (or nearly linear) in the size of the input [FR10,DGGP14,DM15a]. In this work, we initiate the development of sublinear time algorithms that find the planted clique when k=ω(√(n loglog n)). Our algorithms can recover the clique in time O(n+(n/k)^3)=O(n^3/2) when k=Ω(√(nlog n)), and in time O(n^2/(k^2/24n)) for ω(√(nloglog n))=k=o(√(nlogn)). An Ω(n) running time lower bound for the planted clique recovery problem follows easily from the results of [RS19] and therefore our recovery algorithms are optimal whenever k = Ω(n^2/3). As the lower bound of [RS19] builds on purely information theoretic arguments, it cannot provide a detection lower bound stronger than Ω(n^2/k^2). Since our algorithms for k = Ω(√(n log n)) run in time O(n^3/k^3 + n), we show stronger lower bounds based on computational hardness assumptions. With a slightly different notion of the planted clique problem we show that the Planted Clique Conjecture implies the following. A natural family of non-adaptive algorithms—which includes our algorithms for clique detection—cannot reliably solve the planted clique detection problem in time O( n^3-δ/k^3) for any constant δ>0. Thus we provide evidence that if detecting small cliques is hard, it is also likely that detecting large cliques is not too easy.
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