A Poisson Process for Submodular Maximization
For researchers in submodular optimization, this provides a simpler, more elegant algorithm matching the state-of-the-art approximation ratio, though it is an incremental contribution as it does not improve the known bound.
This paper introduces a novel Poisson process-based algorithm for maximizing a monotone submodular function under a matroid constraint, achieving the tight (1-1/e) approximation guarantee with fewer element swaps and without discretization or rounding. The approach also yields fast algorithms for submodular welfare maximization and assignment problems.
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.