On Finding Randomly Planted Cliques in Arbitrary Graphs

TL;DR

This paper introduces a simple, nearly linear time algorithm for detecting large planted cliques in arbitrary graphs with bounded maximum degree, demonstrating a separation from worst-case hardness assumptions.

Contribution

The paper presents a deterministic algorithm that efficiently recovers large planted cliques in graphs with bounded degree, extending techniques beyond the classical models.

Findings

Algorithm recovers cliques of size proportional to n with high probability.

Shows correlation between vertex degrees and planted clique in the final graph.

Extends techniques to planted biclique detection, surpassing worst-case guarantees.

Abstract

We study a planted clique model introduced by Feige where a complete graph of size $c\cdot n$ is planted uniformly at random in an arbitrary $n$-vertex graph. We give a simple deterministic algorithm that, in almost linear time, recovers a clique of size $(c/3)^{O(1/c)} \cdot n$ as long as the original graph has maximum degree at most $(1-p)n$ for some fixed $p>0$. The proof hinges on showing that the degrees of the final graph are correlated with the planted clique, in a way similar to (but more intricate than) the classical $G(n,\frac{1}{2})+K_{\sqrt{n}}$ planted clique model. Our algorithm suggests a separation from the worst-case model, where, assuming the Unique Games Conjecture, no polynomial algorithm can find cliques of size $\Omega(n)$ for every fixed $c>0$, even if the input graph has maximum degree $(1-p)n$. Our techniques extend beyond the planted clique model. For example, when the planted graph is a balanced biclique, we recover a balanced biclique of size larger than the best guarantees known for the worst case.