Planted Bipartite Graph Detection

02/07/2023

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by   Asaf Rotenberg, et al.

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We consider the task of detecting a hidden bipartite subgraph in a given random graph. Specifically, under the null hypothesis, the graph is a realization of an Erdős-Rényi random graph over n vertices with edge density q. Under the alternative, there exists a planted k_𝖱× k_𝖫 bipartite subgraph with edge density p>q. We derive asymptotically tight upper and lower bounds for this detection problem in both the dense regime, where q,p = Θ(1), and the sparse regime where q,p = Θ(n^-α), α∈(0,2]. Moreover, we consider a variant of the above problem, where one can only observe a relatively small part of the graph, by using at most 𝖰 edge queries. For this problem, we derive upper and lower bounds in both the dense and sparse regimes.