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Random graph models with community structure have been studied extensively inthe literature. For both the problems of detecting and recovering communitystructure, an interesting landscape of statistical and computational phasetransitions has emerged. A natural unanswered question is: might it be possibleto infer properties of the community structure (for instance, the number andsizes of communities) even in situations where actually finding thosecommunities is believed to be computationally hard? We show the answer is no.In particular, we consider certain hypothesis testing problems between modelswith different community structures, and we show (in the low-degree polynomialframework) that testing between two options is as hard as finding thecommunities.  In addition, our methods give the first computational lower bounds fortesting between two different `planted' distributions, whereas previous resultshave considered testing between a planted distribution and an i.i.d. `null'distribution.

Is it easier to count communities than find them?