Fast Adaptive Non-Monotone Submodular Maximization Subject to a Knapsack Constraint

TL;DR

This paper introduces a fast, randomized greedy algorithm for non-monotone submodular maximization under a knapsack constraint, achieving strong approximation ratios and efficiency, and extends it to stochastic settings with proven guarantees.

Contribution

The paper presents a novel, efficient randomized greedy algorithm with a 5.83 approximation for non-monotone submodular maximization under knapsack constraints, and extends it to stochastic problems with a 9-approximation.

Findings

Achieves a 5.83 approximation ratio in $O(n \,\log n)$ time.

Extends to stochastic problems with a 9-approximate adaptive policy.

Demonstrates improved empirical performance on real and synthetic data.

Abstract

Constrained submodular maximization problems encompass a wide variety of applications, including personalized recommendation, team formation, and revenue maximization via viral marketing. The massive instances occurring in modern day applications can render existing algorithms prohibitively slow, while frequently, those instances are also inherently stochastic. Focusing on these challenges, we revisit the classic problem of maximizing a (possibly non-monotone) submodular function subject to a knapsack constraint. We present a simple randomized greedy algorithm that achieves a $5.83$ approximation and runs in $O(n \log n)$ time, i.e., at least a factor $n$ faster than other state-of-the-art algorithms. The robustness of our approach allows us to further transfer it to a stochastic version of the problem. There, we obtain a 9-approximation to the best adaptive policy, which is the first constant approximation for non-monotone objectives. Experimental evaluation of our algorithms showcases their improved performance on real and synthetic data.