Finding Planted Cliques in Sublinear Time

We study the planted clique problem in which a clique of size $k$ is planted in an Erd\H{o}s-R\'enyi 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 initiate the development of 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 non-adaptive algorithms---which includes our algorithms for clique detection---cannot 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

Is the Space Complexity of Planted Clique Recovery the Same as That of Detection?

We study the planted clique problem in which a clique of size k is planted in an Erd\H{o}s-R\'enyi graph G(n, 1/2), and one is interested in either detecting or recovering this planted clique. This problem is interesting because it is widely believed to show a statistical-computational gap at clique size k=sqrt{n}, and has emerged as the prototypical problem with such a gap from which average-case hardness of other statistical problems can be deduced. It also displays a tight computational connection between the detection and recovery variants, unlike other problems of a similar nature. This wide investigation into the computational complexity of the planted clique problem has, however, mostly focused on its time complexity. In this work, we ask- Do the statistical-computational phenomena that make the planted clique an interesting problem also hold when we use `space efficiency' as our notion of computational efficiency? It is relatively easy to show that a positive answer to this question depends on the existence of a O(log n) space algorithm that can recover planted cliques of size k = Omega(sqrt{n}). Our main result comes very close to designing such an algorithm. We show that for k=Omega(sqrt{n}), the recovery problem can be solved in O((log*{n}-log*{k/sqrt{n}}) log n) bits of space. 1. If k = omega(sqrt{n}log^{(l)}n) for any constant integer l > 0, the space usage is O(log n) bits. 2.If k = Theta(sqrt{n}), the space usage is O(log*{n} log n) bits. Our result suggests that there does exist an O(log n) space algorithm to recover cliques of size k = Omega(sqrt{n}), since we come very close to achieving such parameters. This provides evidence that the statistical-computational phenomena that (conjecturally) hold for planted clique time complexity also (conjecturally) hold for space complexity

Finding cliques and dense subgraphs using edge queries

We consider the problem of finding a large clique in an Erd\H{o}s--R\'enyi random graph where we are allowed unbounded computational time but can only query a limited number of edges. Recall that the largest clique in $G \sim G(n,1/2)$ has size roughly $2\log_{2} n$. Let $\alpha_{\star}(\delta,\ell)$ be the supremum over $\alpha$ such that there exists an algorithm that makes $n^{\delta}$ queries in total to the adjacency matrix of $G$, in a constant $\ell$ number of rounds, and outputs a clique of size $\alpha \log_{2} n$ with high probability. We give improved upper bounds on $\alpha_{\star}(\delta,\ell)$ for every $\delta \in [1,2)$ and $\ell \geq 3$. We also study analogous questions for finding subgraphs with density at least $\eta$ for a given $\eta$, and prove corresponding impossibility results.Comment: 19 pp, 5 figures, Focused Workshop on Networks and Their Limits held at the Erd\H{o}s Center, Budapest, Hungary in July 202

Optimal Rates of Convergence for Noisy Sparse Phase Retrieval via Thresholded Wirtinger Flow

This paper considers the noisy sparse phase retrieval problem: recovering a sparse signal $x \in \mathbb{R}^p$ from noisy quadratic measurements $y_j = (a_j' x )^2 + \epsilon_j$, $j=1, \ldots, m$, with independent sub-exponential noise $\epsilon_j$. The goals are to understand the effect of the sparsity of $x$ on the estimation precision and to construct a computationally feasible estimator to achieve the optimal rates. Inspired by the Wirtinger Flow [12] proposed for noiseless and non-sparse phase retrieval, a novel thresholded gradient descent algorithm is proposed and it is shown to adaptively achieve the minimax optimal rates of convergence over a wide range of sparsity levels when the $a_j$'s are independent standard Gaussian random vectors, provided that the sample size is sufficiently large compared to the sparsity of $x$.Comment: 28 pages, 4 figure

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum