Linear Query Approximation Algorithms for Non-monotone Submodular Maximization under Knapsack Constraint

TL;DR

This paper introduces two new linear-query algorithms with constant approximation factors for non-monotone submodular maximization under a knapsack constraint, achieving efficient solutions with fewer queries.

Contribution

It presents the first linear-query, constant-factor approximation algorithms for non-monotone submodular maximization under knapsack constraints, combining threshold greedy and set partitioning techniques.

Findings

Algorithms achieve comparable results to state-of-the-art methods.

Significantly reduce the number of queries needed.

Effective in applications like revenue maximization and image summarization.

Abstract

This work, for the first time, introduces two constant factor approximation algorithms with linear query complexity for non-monotone submodular maximization over a ground set of size $n$ subject to a knapsack constraint, $\mathsf{DLA}$ and $\mathsf{RLA}$. $\mathsf{DLA}$ is a deterministic algorithm that provides an approximation factor of $6+\epsilon$ while $\mathsf{RLA}$ is a randomized algorithm with an approximation factor of $4+\epsilon$. Both run in $O(n \log(1/\epsilon)/\epsilon)$ query complexity. The key idea to obtain a constant approximation ratio with linear query lies in: (1) dividing the ground set into two appropriate subsets to find the near-optimal solution over these subsets with linear queries, and (2) combining a threshold greedy with properties of two disjoint sets or a random selection process to improve solution quality. In addition to the theoretical analysis, we have evaluated our proposed solutions with three applications: Revenue Maximization, Image Summarization, and Maximum Weighted Cut, showing that our algorithms not only return comparative results to state-of-the-art algorithms but also require significantly fewer queries.