Fast Monotone Summation over Disjoint Sets

TL;DR

This paper introduces a monotone arithmetic circuit for efficiently computing sums over disjoint subsets, with applications in graph algorithms, matrix permanents, and machine learning feature selection.

Contribution

It presents a novel, subtraction-free circuit design for disjoint subset summation with near-optimal complexity for fixed parameters.

Findings

Achieves $O((n^p+n^q) \, \log n)$ complexity for the problem

Provides applications to counting heaviest paths and matrix permanents

Offers improved algorithms for dynamic feature selection in machine learning

Abstract

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 $O((n^p+n^q)\log n)$ 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 tree-projection 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.