Streaming Submodular Maximization under a $k$-Set System Constraint

TL;DR

This paper introduces a new framework converting streaming algorithms for monotone submodular maximization into those for non-monotone cases, achieving tight approximation ratios under various complex constraints.

Contribution

It presents the first streaming algorithms for monotone submodular maximization under $k$-extendible and $k$-set system constraints, with improved approximation ratios.

Findings

Achieved the tightest deterministic approximation ratio for $k$-matchoid constraints.

Developed $O(k\log k)$ and $O(k^2\log k)$ approximation algorithms for specific constraints.

Extensive experiments demonstrate superior empirical performance across diverse applications.

Abstract

In this paper, we propose a novel framework that converts streaming algorithms for monotone submodular maximization into streaming algorithms for non-monotone submodular maximization. This reduction readily leads to the currently tightest deterministic approximation ratio for submodular maximization subject to a $k$-matchoid constraint. Moreover, we propose the first streaming algorithm for monotone submodular maximization subject to $k$-extendible and $k$-set system constraints. Together with our proposed reduction, we obtain $O(k\log k)$ and $O(k^2\log k)$ approximation ratio for submodular maximization subject to the above constraints, respectively. We extensively evaluate the empirical performance of our algorithm against the existing work in a series of experiments including finding the maximum independent set in randomly generated graphs, maximizing linear functions over social networks, movie recommendation, Yelp location summarization, and Twitter data summarization.