Finding cliques using few probes

TL;DR

This paper proves that for random graphs, any algorithm with limited probes and rounds is unlikely to find large cliques, highlighting fundamental limits of clique detection under resource constraints.

Contribution

It establishes lower bounds on the size of cliques that algorithms with few probes and rounds can find in random graphs.

Findings

Algorithms with fewer than $n^{ ext{delta}}$ probes and constant rounds cannot reliably find large cliques.

The maximum clique size found by such algorithms is bounded by a constant times $ ext{log} n$.

The results apply to random graphs from $G_{n,1/2}$, which typically contain cliques of size about $2 ext{log} n$.

Abstract

Consider algorithms with unbounded computation time that probe the entries of the adjacency matrix of an $n$ vertex graph, and need to output a clique. We show that if the input graph is drawn at random from $G_{n,\frac{1}{2}}$ (and hence is likely to have a clique of size roughly $2\log n$), then for every $\delta < 2$ and constant $\ell$, there is an $\alpha < 2$ (that may depend on $\delta$ and $\ell$) such that no algorithm that makes $n^{\delta}$ probes in $\ell$ rounds is likely (over the choice of the random graph) to output a clique of size larger than $\alpha \log n$.