Fast Monotone Summation over Disjoint Sets

We study the problem of computing an ensemble of multiple sums where the summands in each sum are indexed by subsets of size p of an n-element ground set. More precisely, the task is to compute, for each subset of size q of the ground set, the sum over the values of all subsets of size p that are disjoint from the subset of size q. We present an arithmetic circuit that, without subtraction, solves the problem using O((n + n) log n) arithmetic gates, all monotone; for constant p, q this is within the factor log n of the optimal. The circuit design is based on viewing the summation as a “set nucleation” task and using a treeprojection approach to implement the nucleation. Applications include improved algorithms for counting heaviest k-paths in a weighted graph, computing permanents of rectangular matrices, and dynamic feature selection in machine learning.