Optimal Streaming Algorithms for Submodular Maximization with Cardinality Constraints

TL;DR

This paper introduces a deterministic, single-pass streaming algorithm for non-monotone submodular maximization under a cardinality constraint, achieving near-optimal approximation guarantees with efficient memory and update time.

Contribution

It presents a novel deterministic streaming algorithm that combines with offline post-processing to improve approximation guarantees for submodular maximization.

Findings

Achieves a 0.2779-approximation with polynomial time using a state-of-the-art offline algorithm.

Uses roughly O(k / ε^2) memory and O((log k + log(1/α))/ε^2) update time per element.

Improves previous polynomial-time approximation from 0.1715 to 0.2779.

Abstract

We study the problem of maximizing a non-monotone submodular function subject to a cardinality constraint in the streaming model. Our main contribution is a single-pass (semi-)streaming algorithm that uses roughly $O(k / \varepsilon^2)$ memory, where $k$ is the size constraint. At the end of the stream, our algorithm post-processes its data structure using any offline algorithm for submodular maximization, and obtains a solution whose approximation guarantee is $\frac{\alpha}{1+\alpha}-\varepsilon$, where $\alpha$ is the approximation of the offline algorithm. If we use an exact (exponential time) post-processing algorithm, this leads to $\frac{1}{2}-\varepsilon$ approximation (which is nearly optimal). If we post-process with the algorithm of Buchbinder and Feldman (Math of OR 2019), that achieves the state-of-the-art offline approximation guarantee of $\alpha=0.385$, we obtain $0.2779$-approximation in polynomial time, improving over the previously best polynomial-time approximation of $0.1715$ due to Feldman et al. (NeurIPS 2018). It is also worth mentioning that our algorithm is combinatorial and deterministic, which is rare for an algorithm for non-monotone submodular maximization, and enjoys a fast update time of $O(\frac{\log k + \log (1/\alpha)}{\varepsilon^2})$ per element.