Non-Monotone Submodular Maximization with Multiple Knapsacks in Static and Dynamic Settings

TL;DR

This paper introduces a simple greedy algorithm for maximizing non-monotone submodular functions under multiple knapsack constraints, providing strong approximation guarantees in both static and dynamic settings, with applications demonstrated in video summarization and sensor placement.

Contribution

It presents a novel discrete greedy algorithm with proven approximation guarantees that does not require problem relaxation, effective in static and dynamic environments, and applicable to real-world tasks.

Findings

The greedy algorithm achieves strong approximation guarantees for bounded curvature functions.

In dynamic settings, the modified greedy avoids full restarts and maintains solution quality.

Experimental results show competitive performance in video summarization and sensor placement tasks.

Abstract

We study the problem of maximizing a non-monotone submodular function under multiple knapsack constraints. We propose a simple discrete greedy algorithm to approach this problem, and prove that it yields strong approximation guarantees for functions with bounded curvature. In contrast to other heuristics, this requires no problem relaxation to continuous domains and it maintains a constant-factor approximation guarantee in the problem size. In the case of a single knapsack, our analysis suggests that the standard greedy can be used in non-monotone settings. Additionally, we study this problem in a dynamic setting, by which knapsacks change during the optimization process. We modify our greedy algorithm to avoid a complete restart at each constraint update. This modification retains the approximation guarantees of the static case. We evaluate our results experimentally on a video summarization and sensor placement task. We show that our proposed algorithm competes with the state-of-the-art in static settings. Furthermore, we show that in dynamic settings with tight computational time budget, our modified greedy yields significant improvements over starting the greedy from scratch, in terms of the solution quality achieved.