# Finding Planted Cliques in Sublinear Time

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.

READ FULL TEXT
Comments

There are no comments yet.