  Consider a collection of weighted subsets of a ground set N. Given a query
subset Q of N, how fast can one (1) find the weighted sum over all subsets of
Q, and (2) sample a subset of Q proportionally to the weights? We present a
tree-based greedy heuristic, Treedy, that for a given positive tolerance d
answers such counting and sampling queries to within a guaranteed relative
error d and total variation distance d, respectively. Experimental results on
artificial instances and in application to Bayesian structure discovery in
Bayesian networks show that approximations yield dramatic savings in running
time compared to exact computation, and that Treedy typically outperforms a
previously proposed sorting-based heuristic.
