For the core variable selection problem under the Hamming loss, we derive a non-asymptotic exact minimax selector over the class of all s-sparse vectors, which is also the Bayes selector with respect to the uniform prior. While this optimal selector is, in general, not realizable in polynomial time, we show that its tractable counterpart (the scan selector) attains the minimax expected Hamming risk to within factor 2. Moreover it is non-asymptotically exact minimax under the probability of wrong recovery criterion. In the monotone likelihood ratio framework, we establish explicit lower bounds on the minimax risk and provide its tight characterization in terms of the best separable selector risk. As a consequence, we obtain sharp necessary and sufficient conditions of exact and almost full recovery in the location model with light tail distributions and in the problem of group variable selection under Gaussian noise. Joint work with Cristina Butucea, Enno Mammen and Simo Ndaoud.