Detecting communities is Hard (And Counting Them is Even Harder)

We consider the algorithmic problem of community detection in networks. Given an undirected friendship graph G = (V, E), a subset S ⊆ V is an (α, β)-community if: • Every member of the community is friends with an α-fraction of the community; • Every non-member is friends with at most a β-fraction of the community. Arora et al [AGSS12] gave a quasi-polynomial time algorithm for enumerating all the (α, β)-communities for any constants α > β. Here, we prove that, assuming the Exponential Time Hypothesis (ETH), quasi-polynomial time is in fact necessary-and even for a much weaker approximation desider-atum. Namely, distinguishing between: • G contains an (1, o (1))-community; and • G does not contain an (β + o (1) , β)-community for any β ∈ [0, 1]. We also prove that counting the number of (1, o (1))-communities requires quasi-polynomial time assuming the weaker #ETH.