If it is Good Then Drop it -- a Spiteful Poisson Process for Submodular Maximization

2026-08-05

AI Summary

This work addresses the problem of maximizing general (non-monotone) submodular functions subject to matroid constraints. To this end, the authors propose a hybrid algorithm grounded in a continuous-time Poisson process, which performs single-element exchanges at random time points and introduces an innovative “adversarial discard” mechanism—deliberately removing elements common to both the current solution and the optimal response set. This technique overcomes the limitation of prior methods that only apply to monotone settings. By uniquely integrating adversarial discarding with a Poisson-driven continuous-time framework, the approach unifies the treatment of both monotone and non-monotone submodular objectives. It achieves approximation ratios of $1 - 1/e$ for general matroids and $1/e$ for partition matroids, matching or nearly matching the theoretical optima for these settings.

Abstract

We study the problem of maximizing a general and not necessarily monotone submodular function subject to a matroid independence constraint. This problem has a rich history, with multiple algorithms using both discrete and continuous methods. Recently, [Ganz-Rozenman, Kulik, Schwartz and Singh STOC `26] presented a novel hybrid approach based on a Poisson process that aims to combine the strengths of both discrete and continuous methods for the special case of the problem where the submodular function is monotone. Our main result is a new Poisson process based hybrid algorithm that works for both non-monotone and monotone submodular functions, achieving an approximation of $ \frac{1}{e}$ for the former and $1-\frac{1}{e}$ for the latter. The algorithm always maintains a feasible set and at random times governed by the Poisson process it performs a single element swap based on a best response set. The new idea is that our algorithm is spiteful as it can purposefully discard an element that is in both the current set and the best response set. Surprisingly, this spiteful step does not harm the approximation our algorithm achieves for monotone submodular functions but is necessary for the non-monotone case. As applications, we obtain fast approximation algorithms for maximizing non-monotone submodular function subject to a general matroid independence constraint as well as faster algorithms for a partition matroid.

Problem

• submodular maximization
• matroid constraint
• non-monotone submodular function
• approximation algorithm

Innovation

• spiteful Poisson process
• submodular maximization
• non-monotone submodular functions
• matroid constraint
• hybrid algorithm