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Reconstruction of population histories is a central problem in populationgenetics. Existing coalescent-based methods, like the seminal work of Li andDurbin (Nature, 2011), attempt to solve this problem using sequence data buthave no rigorous guarantees. Determining the amount of data needed to correctlyreconstruct population histories is a major challenge. Using a variety of toolsfrom information theory, the theory of extremal polynomials, and approximationtheory, we prove new sharp information-theoretic lower bounds on the problem ofreconstructing population structure -- the history of multiple subpopulationsthat merge, split and change sizes over time. Our lower bounds are exponentialin the number of subpopulations, even when reconstructing recent histories. Wedemonstrate the sharpness of our lower bounds by providing algorithms fordistinguishing and learning population histories with matching dependence onthe number of subpopulations. Along the way and of independent interest, weessentially determine the optimal number of samples needed to learn anexponential mixture distribution information-theoretically, proving the upperbound by analyzing natural (and efficient) algorithms for this problem.

How Many Subpopulations is Too Many? Exponential Lower Bounds for Inferring Population Histories