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 $G=\left(V,E\right)$, a subset $S\subseteq V$ is an $\left(\alpha,\beta\right)$-community if: * Every member of the community is friends with an $\alpha$-fraction of the community; * Every non-member is friends with at most a $\beta$-fraction of the community. Arora et al [AGSS12] gave a quasi-polynomial time algorithm for enumerating all the $\left(\alpha,\beta\right)$-communities for any constants $\alpha>\beta$. Here, we prove that, assuming the Exponential Time Hypothesis (ETH), quasi-polynomial time is in fact necessary - and even for a much weaker approximation desideratum. Namely, distinguishing between: * $G$ contains an $\left(1,o\left(1\right)\right)$-community; and * $G$ does not contain an $\left(\beta+o\left(1\right),\beta\right)$-community for any $\beta\in\left[0,1\right]$. We also prove that counting the number of $\left(1,o\left(1\right)\right)$-communities requires quasi-polynomial time assuming the weaker #ETH.