Submodular Maximization under the Intersection of Matroid and Knapsack Constraints

TL;DR

This paper introduces SPROUT and SPROUT++ algorithms for maximizing submodular functions under combined matroid and knapsack constraints, offering improved approximation guarantees and practical efficiency demonstrated through experiments.

Contribution

It presents the first polynomial-time algorithms with better approximation guarantees for submodular maximization under intersecting matroid and knapsack constraints.

Findings

SPROUT achieves improved approximation guarantees over existing methods.

SPROUT++ maintains similar guarantees with enhanced efficiency.

Experimental results show SPROUT++ outperforms baselines in recommendation and max-cut tasks.

Abstract

Submodular maximization arises in many applications, and has attracted a lot of research attentions from various areas such as artificial intelligence, finance and operations research. Previous studies mainly consider only one kind of constraint, while many real-world problems often involve several constraints. In this paper, we consider the problem of submodular maximization under the intersection of two commonly used constraints, i.e., $k$-matroid constraint and $m$-knapsack constraint, and propose a new algorithm SPROUT by incorporating partial enumeration into the simultaneous greedy framework. We prove that SPROUT can achieve a polynomial-time approximation guarantee better than the state-of-the-art algorithms. Then, we introduce the random enumeration and smooth techniques into SPROUT to improve its efficiency, resulting in the SPROUT++ algorithm, which can keep a similar approximation guarantee. Experiments on the applications of movie recommendation and weighted max-cut demonstrate the superiority of SPROUT++ in practice.