Optimization algorithms for Carleson and sparse collections of sets

TL;DR

This paper introduces optimization-based algorithms to compute Carleson constants and establish sparseness of collections of sets, advancing computational methods in dyadic harmonic analysis.

Contribution

It presents a polynomial algorithm for Carleson constant computation and a constructive proof linking Carleson collections to sparseness using flow-cut duality.

Findings

Polynomial algorithm for Carleson constant calculation

Constructive proof that Carleson collections are sparse

Optimal dependence of constants achieved

Abstract

Carleson and sparse collections of sets play a central role in dyadic harmonic analysis. We employ methods from optimization theory to study such collections. First, we present a strongly polynomial algorithm to compute the Carleson constant of a collection of sets, improving on the recent approximation algorithm of Rey. Our algorithm is based on submodular function minimization. Second, we provide an algorithm showing that any Carleson collection is sparse, achieving optimal dependence of the respective constants and thus providing a constructive proof of a result of H\"anninen. Our key insight is a reformulation of the duality between the Carleson condition and sparseness in terms of the duality between the maximum flow and the minimum cut in a weighted directed graph.