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

Treedy: A Heuristic for Counting and Sampling Subsets