Treedy: A Heuristic for Counting and Sampling Subsets

Consider a collection of weighted subsets of a ground set N. We present a tree-based greedy heuristic, Treedy, that for a given query subset Q of N and a tolerance d approximates the weighted sum over all subsets of Q within relative error d. It also enables approximate sampling of subset of Q proportionally to the weights within total variation distance d. Experimental results show that approximations yield dramatic savings in running time compared to exact computation, and that Treedy typically outperforms a previously proposed sorting-based heuristic.