Finding cliques and dense subgraphs using edge queries

TL;DR

This paper investigates the limits of algorithms that find large cliques and dense subgraphs in Erdős–Rényi graphs using a limited number of edge queries, providing new upper bounds on achievable clique sizes.

Contribution

It introduces improved upper bounds on the size of cliques that can be found with limited queries and rounds, advancing understanding of query complexity in random graph problems.

Findings

Derived upper bounds on clique size with limited queries and rounds.

Established impossibility results for finding dense subgraphs with high probability.

Abstract

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.