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Title: Random Subgraph Detection Using Queries
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 Erdös-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.  more » « less
Award ID(s):
2217058 2133484
PAR ID:
10520716
Author(s) / Creator(s):
; ;
Publisher / Repository:
JMLR
Date Published:
Journal Name:
Journal of machine learning research
ISSN:
1532-4435
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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