Finding Planted Cliques in Sublinear Time
Abstract
We study the planted clique problem in which a clique of size $k$ is planted in an ErdősRényi graph of size $n$ and one wants to recover this planted clique. For $k=\Omega(\sqrt{n})$, 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 develop sublinear time algorithms that find the planted clique when $k=\omega(\sqrt{n \log \log n})$. Our algorithms can recover the clique in time $\widetilde{O}\left(n+(\frac{n}{k})^{3}\right)=\widetilde{O}\left(n^{\frac{3}{2}}\right)$ when $k=\Omega(\sqrt{n\log n})$, and in time $\widetilde{O}\left(n^2/\exp{\left(\frac{k^2}{24n}\right)}\right)$ for $\omega(\sqrt{n\log \log n})=k=o(\sqrt{n\log{n}})$. An ${\Omega}(n)$ 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 $k = \Omega(n^{\frac{2}{3}})$. As the lower bound of [RS19] builds on purely information theoretic arguments, it cannot provide a detection lower bound stronger than $\widetilde{\Omega}(\frac{n^2}{k^2})$. Since our algorithms for $k = \Omega(\sqrt{n \log n})$ run in time $\widetilde{O}\left(\frac{n^3}{k^3} + n\right)$, 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 nonadaptive algorithmswhich includes our algorithms for clique detectioncannot reliably solve the planted clique detection problem in time $O\left( \frac{n^{3\delta}}{k^3}\right)$ for any constant $\delta>0$. Thus we provide evidence that if detecting small cliques is hard, it is also likely that detecting large cliques is not \textit{too} easy.
 Publication:

arXiv eprints
 Pub Date:
 April 2020
 arXiv:
 arXiv:2004.12002
 Bibcode:
 2020arXiv200412002M
 Keywords:

 Computer Science  Computational Complexity;
 Computer Science  Data Structures and Algorithms;
 Computer Science  Information Theory