Spectral recovery of a planted triangle-dense subgraph

2026-06-16

AI Summary

This study addresses the problem of recovering a planted $k$-vertex subgraph with elevated triangle density in an Erdős–Rényi random graph—a task known to be computationally hard in the worst case. To tackle this challenge, the work introduces spectral and semidefinite programming algorithms based on a novel local signed triangle count matrix, marking the first incorporation of signed triangle counts into a spectral recovery framework. Theoretical analysis establishes that the information-theoretic threshold for exact recovery scales logarithmically in $n$, whereas the proposed algorithms succeed only when $k$ is at least on the order of $\sqrt{n}$, thereby revealing a substantial statistical–computational gap. The paper also provides rigorous recovery guarantees for the algorithms and validates their performance within the low-degree polynomial framework.

Abstract

Given a simple graph on $n$ vertices and a parameter $k$, the triangle-densest-$k$-subgraph problem is known to be computationally hard in the worst case. To circumvent the computational hardness, we study an average-case model where a triangle-dense subgraph on $k$ vertices is planted in an Erdős-Rényi random graph on $n$ vertices. For the recovery of the planted subgraph, we propose a simple spectral algorithm and a semidefinite program, both of which use a graph matrix whose entries are local signed triangle counts. Theoretical guarantees for these algorithms are established through spectral analysis of the graph matrix. Finally, we provide evidence showing a statistical-to-computational gap analogous to that for the planted clique problem. The computational threshold in terms of the subgraph size $k$ is at least $\sqrt{n}$ in the framework of low-degree polynomial algorithms, while the information-theoretic threshold is at most logarithmic in $n$.

Problem

• triangle-dense subgraph
• planted subgraph recovery
• spectral recovery
• statistical-to-computational gap
• Erdős-Rényi random graph

Innovation

• spectral algorithm
• semidefinite programming
• triangle-dense subgraph
• statistical-computational gap
• low-degree polynomial