On Finding Randomly Planted Cliques in Arbitrary Graphs

2025-05-10

AI Summary

This paper addresses the detection and recovery of a planted clique of size $c n$ in arbitrary graphs, focusing on the Feige planting model and graphs with maximum degree at most $(1-p)n$. We propose a deterministic algorithm based on vertex degree distribution that recovers the planted clique in nearly linear time—marking the first worst-case guarantee for non-random graphs. We establish a strong probabilistic correlation between the planted clique and the degree distribution, and extend our framework to planted bicliques and balanced bipartite subgraphs. Our main contribution is a provably correct recovery algorithm for cliques of size $(c/3)^{O(1/c)} n$, significantly improving upon the classical worst-case lower bound of $Omega(sqrt{n})$. This yields the first nontrivial, polynomial-time recovery guarantee with rigorous theoretical justification for arbitrary host graphs under the Feige model.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study a planted clique model introduced by Feige where a complete graph of size $ccdot n$ 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.

Problem

• Finding planted cliques in arbitrary graphs efficiently
• Recovering cliques despite high-degree noise constraints
• Extending techniques beyond cliques to biclique recovery

Innovation

• Deterministic algorithm for planted clique recovery
• Almost linear time complexity
• Handles arbitrary graphs with degree constraints