A Poisson Process for Submodular Maximization

TL;DR

This paper introduces a simple, Poisson process-based method for maximizing monotone submodular functions under matroid constraints, achieving the optimal approximation ratio with fewer swaps and no discretization.

Contribution

It presents a novel Poisson process approach that matches the best known approximation guarantees without discretization or rounding, simplifying the algorithmic process.

Findings

Achieves the tight (1 - 1/e) approximation ratio.

Requires fewer element swaps compared to previous methods.

Provides fast algorithms for submodular welfare and assignment problems.

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{\'a}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.