A tight bound for the clique query problem in two rounds

TL;DR

This paper establishes a tight upper bound for the size of large cliques that can be found in a random graph using a limited number of adaptive query rounds, advancing understanding of query complexity in clique detection.

Contribution

It provides a tight bound for the two-round query model in the clique problem, improving the theoretical limits on clique size detection in random graphs.

Findings

Two-round algorithms cannot find cliques larger than (4/3)δ log n for certain parameters.

Improved upper bounds for multi-round algorithms in specific query regimes.

Tight bounds are proven when early rounds have fewer queries than the last round.

Abstract

We consider a problem introduced by Feige, Gamarnik, Neeman, R\'acz and Tetali [2020], that of finding a large clique in a random graph $G\sim G(n,\frac{1}{2})$, where the graph $G$ is accessible by queries to entries of its adjacency matrix. The query model allows some limited adaptivity, with a constant number of rounds of queries, and $n^\delta$ queries in each round. With high probability, the maximum clique in $G$ is of size roughly $2\log n$, and the goal is to find cliques of size $\alpha\log n$, for $\alpha$ as large as possible. We prove that no two-rounds algorithm is likely to find a clique larger than $\frac{4}{3}\delta\log n$, which is a tight upper bound when $1\leq\delta\leq \frac{6}{5}$. For other ranges of parameters, namely, two-rounds with $\frac{6}{5}<\delta<2$, and three-rounds with $1\leq\delta<2$, we improve over the previously known upper bounds on $\alpha$, but our upper bounds are not tight. If early rounds are restricted to have fewer queries than the last round, then for some such restrictions we do prove tight upper bounds.