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

TL;DR

This work addresses maximizing a possibly non-monotone submodular function under a knapsack constraint, focusing on massive and stochastic instances. It introduces SampleGreedy, a simple randomized density-greedy framework that achieves a $5.83$-approximation in $O(n \log n)$ value oracle calls, with a lazy variant attaining $3+2\sqrt{2}+\varepsilon$ while maintaining nearly-linear complexity; it also extends to adaptive non-monotone submodular maximization, yielding a $9$-approximation to the best adaptive policy. The adaptive extension, AdaptiveGreedy, achieves the first constant-factor guarantee for non-monotone objectives in the adaptive submodular setting, with a lazy version giving $9+\varepsilon$. Empirical results on video recommendation and influence-exploit tasks corroborate the theoretical findings, showing SampleGreedy consistently outperforms FANTOM in both value and efficiency across large-scale datasets and budgets, highlighting the approach’s practicality and scalability.

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.

Figures (1)

• Figure :

Theorems & Definitions (23)

• Definition 1: Submodularity
• Theorem 1: NemhauserWF78
• Lemma 1: Lemma 2.2. of BuchbinderFNS14
• Definition 2
• Theorem 2
• proof
• Lemma 2
• proof : Proof of \ref{['lem:O_vs_SUS*']}
• Lemma 3
• proof : Proof of \ref{['lem:S_vs_sum']}