Fundamental Limits of Query-Based Subgraph Detection

2026-07-19

AI Summary

This work addresses the problem of efficiently detecting an arbitrarily structured subgraph embedded in a random graph under the constraint of only a limited number of non-adaptive edge queries. By establishing matching information-theoretic lower bounds and algorithmic upper bounds, the paper presents the first unified framework for query complexity applicable to general embedded subgraphs, extending classical full-observation models to information-constrained settings. Leveraging structural properties such as local dense motifs, high-degree vertices, and global edge density, the authors design detection algorithms whose query complexities are tight up to polylogarithmic factors across broad classes of subgraphs—including clique-like, bounded-cover, and hub-dominated structures—thereby achieving near-optimal performance in terms of query efficiency.

Abstract

The planted subgraph detection problem asks whether a random graph contains a hidden structured subgraph. In the classical formulation, the entire adjacency matrix is observed and one distinguishes between an Erdős--Rényi random graph and one obtained by planting a copy of a prescribed graph inside an Erdős--Rényi random graph. The statistical and computational limits of this problem under full observation are now well understood, even for arbitrary planted subgraphs. In this paper, we investigate an information-limited version of the problem in which the planted structure is an arbitrary sequence of graphs $Γ=(Γ_n)_{n\geq1}$, where $Γ_n$ is embedded in an ambient graph on $n$ vertices, but the observer does not have access to the full adjacency matrix. Instead, information is acquired through a limited number of non-adaptive edge queries. We study the minimum query complexity required for reliable detection. We derive general information-theoretic lower bounds and complementary algorithmic upper bounds on the query complexity as functions of the query budget and structural properties of the planted graph. The proposed algorithms exploit three distinct structural mechanisms: dense local motifs, high-degree vertices, and global edge density. We establish matching bounds, up to polylogarithmic factors, for several broad families of planted graphs, including clique-like, bounded-cover, and hub-dominated graph classes. Our framework substantially generalizes existing query-complexity results for planted clique and planted dense subgraph models, providing a unified treatment of arbitrary planted subgraphs under restricted graph access.

Problem

• subgraph detection
• query complexity
• planted subgraph
• non-adaptive queries
• information-theoretic limits

Innovation

• query complexity
• planted subgraph detection
• non-adaptive edge queries
• information-theoretic lower bounds
• structural graph properties