Random Subgraph Detection Using Queries

TL;DR

This paper investigates the problem of detecting a dense subgraph within a larger graph using a limited number of adaptive edge queries, providing bounds on query complexity and algorithms for detection.

Contribution

It introduces a query-based model for subgraph detection, determining the necessary and sufficient number of queries, and proposes algorithms with different computational complexities.

Findings

Established query bounds for detection

Proposed a quasi-polynomial time optimal algorithm

Designed a polynomial-time detection algorithm with more queries

Abstract

The planted densest subgraph detection problem refers to the task of testing whether in a given (random) graph there is a subgraph that is unusually dense. Specifically, we observe an undirected and unweighted graph on $n$ vertices. Under the null hypothesis, the graph is a realization of an Erd\H{o}s-R\'{e}nyi graph with edge probability (or, density) $q$. Under the alternative, there is a subgraph on $k$ vertices with edge probability $p>q$. The statistical as well as the computational barriers of this problem are well-understood for a wide range of the edge parameters $p$ and $q$. In this paper, we consider a natural variant of the above problem, where one can only observe a relatively small part of the graph using adaptive edge queries. For this model, we determine the number of queries necessary and sufficient (accompanied with a quasi-polynomial optimal algorithm) for detecting the presence of the planted subgraph. We also propose a polynomial-time algorithm which is able to detect the planted subgraph, albeit with more queries compared to the above lower bound. We conjecture that in the leftover regime, no polynomial-time algorithms exist. Our results resolve two open questions posed in the past literature.