A Poisson Process for Submodular Maximization

2026-05-04

AI Summary

This work addresses the problem of maximizing a monotone submodular function subject to a matroid independence constraint. The authors introduce, for the first time, a stochastic Poisson process into this domain and propose a novel algorithm that achieves efficient optimization through only a small number of single-element exchanges, without requiring discretization or rounding. The method features a simple structure, circumventing the need for complex rounding procedures inherent in traditional approaches, while attaining the tight $(1-1/e)$ approximation guarantee. As applications, the framework effectively solves submodular welfare maximization as well as general and separable assignment problems, yielding significant improvements in computational efficiency.

Abstract

We study the problem of maximizing a monotone submodular function subject to a matroid independence constraint. For more than a decade, a rich body of work has studied this problem. Initially, a tight approximation of $ (1-\frac{1}{e})$ was given using the continuous greedy algorithm [Calinescu-Chekuri-Pal-Vondr{á}k STOC`2008] and later non-oblivious local search techniques were able to match this tight approximation guarantee [Filmus-Ward FOCS`2012] and [Buchbinder-Feldman FOCS`2024]. We propose a new and remarkably simple approach to this problem that is based on a stochastic Poisson process. Our approach matches the tight $ (1-\frac{1}{e})$ approximation guarantee and it differs from the known two techniques since it does not require discretization or rounding while performing very few single element swaps. We also present applications of our approach and obtain fast algorithms for submodular welfare maximization, and for the general and separable assignment problems.

Problem

• submodular maximization
• matroid constraint
• monotone submodular function
• combinatorial optimization

Innovation

• Poisson process
• submodular maximization
• matroid constraint
• approximation algorithm
• stochastic optimization