# Finding a planted clique by adaptive probing

## TL;DR

This paper investigates the query complexity of detecting and finding a planted clique in a random graph, establishing bounds on the number of adaptive edge queries needed for both tasks.

## Contribution

It provides nearly tight bounds on the number of adaptive queries required for detection and finding of planted cliques, highlighting the query complexity in this problem.

## Key findings

- Detection requires roughly n^2 / k^2 queries.
- Finding the clique requires roughly (n^2 / k^2) log^2 n + n log n queries.
- No algorithms with fewer than o(n^2 / k^2 + n) queries can reliably find the clique.

## Abstract

We consider a variant of the planted clique problem where we are allowed unbounded computational time but can only investigate a small part of the graph by adaptive edge queries. We determine (up to logarithmic factors) the number of queries necessary both for detecting the presence of a planted clique and for finding the planted clique. Specifically, let $G \sim G(n,1/2,k)$ be a random graph on $n$ vertices with a planted clique of size $k$. We show that no algorithm that makes at most $q = o(n^2 / k^2 + n)$ adaptive queries to the adjacency matrix of $G$ is likely to find the planted clique. On the other hand, when $k \geq (2+\epsilon) \log_2 n$ there exists a simple algorithm (with unbounded computational power) that finds the planted clique with high probability by making $q = O( (n^2 / k^2) \log^2 n + n \log n)$ adaptive queries. For detection, the additive $n$ term is not necessary: the number of queries needed to detect the presence of a planted clique is $n^2 / k^2$ (up to logarithmic factors).