[Paper Review] Finding Planted Cliques in Sublinear Time

Jay Mardia, Hilal Asi+1 more|arXiv (Cornell University)|Apr 24, 2020

Complexity and Algorithms in Graphs30 references4 citations

TL;DR

This paper presents simple sublinear-time algorithms for recovering a planted clique in an Erdős-Rényi random graph, leveraging the clique-completion property to achieve runtime as low as $\widetilde{O}(n^{3/2})$ for $k = \Theta(\sqrt{n \log n})$. It establishes that recovery is possible without examining the entire graph when $k = \omega(\sqrt{n})$, and proves optimality of these algorithms for $k = \Omega(n^{2/3})$ using information-theoretic lower bounds.

ABSTRACT

We study the planted clique problem in which a clique of size k is planted in an Erdos-Renyi graph G(n,1/2) and one is interested in recovering this planted clique. It is widely believed that it exhibits a statistical-computational gap when computational efficiency is equated with the existence of polynomial time algorithms. We study this problem under a more fine-grained computational lens and consider the following two questions. 1. Do there exist sublinear time algorithms for recovering the planted clique? 2. What is the smallest running time any algorithm can hope to have? We show that because of a well known clique-completion property, very elementary sublinear time recovery algorithms do indeed exist for clique sizes k = ω(\sqrt{n}). This points to a qualitatively stronger statistical-computational gap. The planted clique recovery problem can be solved without even looking at most of the input above the Θ(\sqrt{n}) threshold and cannot be solved by any efficient algorithm below it. A running time lower bound for the recovery problem follows easily from the results of [RS19], and this implies our recovery algorithms are optimal whenever k = Ω(n^{2/3}). However, for k = o(n^{2/3}) there is a gap between our algorithmic upper bound and the information-theoretic lower bound implied by [RS19]. With some caveats, we show stronger detection lower bounds based on the Planted Clique Conjecture for a natural but restricted class of algorithms. The key idea is to relate very fast sublinear time algorithms for detecting large planted cliques to polynomial time algorithms for detecting small planted cliques.

Motivation & Objective

• To investigate whether sublinear-time algorithms can recover a planted clique in an Erdős-Rényi graph $G(n, \frac{1}{2})$.
• To determine the minimal running time any algorithm can achieve for recovering a planted clique.
• To explore whether the statistical-computational gap in the planted clique problem is qualitatively larger than previously thought when considering sublinear-time algorithms.
• To establish lower bounds for recovery based on the Planted Clique Conjecture, particularly for large clique sizes.

Proposed method

• Proposes Algorithm 2, which runs in $\widetilde{O}(n^{3/2})$ time and recovers the clique with high probability when $k = \Theta(\sqrt{n \log n})$.
• Introduces Algorithm 3, an extension that achieves $\widetilde{O}\left(\left(\frac{n}{k}\right)^3 + n\right)$ runtime for larger cliques.
• Develops Algorithm 4 with runtime $O\left(\frac{n^2}{\frac{k^2}{n} \exp\left(\frac{k^2}{24n}\right)}\right)$ for $k = \omega(\sqrt{n})$ and $k = o(\sqrt{n \log n})$.
• Uses the clique-completion property as the core mechanism: if a subset of the clique is sampled, the rest can be recovered via neighborhood intersection.
• Applies Chernoff bounds and subsampling concentration lemmas to control failure probabilities in random sampling steps.
• Employs genie-aided analysis and conditional probability bounds to show that the clique-completion step succeeds with high probability when the sampled set contains a sufficient fraction of the planted clique.

Experimental results

Research questions

• RQ1 Do sublinear-time algorithms exist for recovering a planted clique in $G(n, \frac{1}{2})$ when $k = \omega(\sqrt{n})$?
• RQ2 What is the smallest running time any algorithm can achieve for recovering a planted clique, and is this bound tight?
• RQ3 Can the Planted Clique Conjecture be used to derive stronger lower bounds for large planted cliques, even when the clique is large?
• RQ4 Is there a fine-grained computational complexity separation between detection and recovery in the planted clique problem for large $k$?
• RQ5 Can the clique-completion property be exploited to design efficient sublinear-time recovery algorithms?

Key findings

• Algorithm 2 runs in $\widetilde{O}(n^{3/2})$ time and recovers the planted clique with high probability when $k = \Theta(\sqrt{n \log n})$.
• For $k = \Omega(n^{2/3})$, the proposed algorithms achieve the information-theoretic lower bound of $\Omega\left(\frac{n^2}{k^2} + n\right)$, proving their optimality.
• The runtime of the recovery algorithm stabilizes at $\widetilde{\Theta}(n)$ for $k = \Omega(n^{2/3})$, meaning further increasing $k$ does not make recovery faster.
• Detection algorithms run in $\widetilde{O}\left(\frac{n^3}{k^3}\right)$ time, showing a fine-grained separation between detection and recovery for large $k$, despite known reductions from recovery to detection.
• The failure probability of Algorithm 4 is bounded by $\frac{1}{3} + o(1)$, with the remaining error probability $o(1)$ controlled via concentration bounds and genie-aided analysis.
• The analysis confirms that for $k = \omega(\sqrt{n})$, the planted clique can be recovered without examining the full graph, highlighting a qualitatively stronger statistical-computational gap.