  Consider a population consisting of n individuals, each of whom has one of d
types (e.g. their blood type, in which case d=4). We are allowed to query this
database by specifying a subset of the population, and in response we observe a
noiseless histogram (a d-dimensional vector of counts) of types of the pooled
individuals. This measurement model arises in practical situations such as
pooling of genetic data and may also be motivated by privacy considerations. We
are interested in the number of queries one needs to unambiguously determine
the type of each individual. In this paper, we study this information-theoretic
question under the random, dense setting where in each query, a random subset
of individuals of size proportional to n is chosen. This makes the problem a
particular example of a random constraint satisfaction problem (CSP) with a
"planted" solution. We establish almost matching upper and lower bounds on the
minimum number of queries m such that there is no solution other than the
planted one with probability tending to 1 as n tends to infinity. Our proof
relies on the computation of the exact "annealed free energy" of this model in
the thermodynamic limit, which corresponds to the exponential rate of decay of
the expected number of solution to this planted CSP. As a by-product of the
analysis, we show an identity of independent interest relating the Gaussian
integral over the space of Eulerian flows of a graph to its spanning tree
polynomial.
