Abstract
Submodular maximization is an important building block for developing algorithms in many areas such as machine learning and data mining. Due to the NP-hardness of the problem, analysis of submodular maximization algorithms typically provides pessimistic worst-case approximation factors only. It is not easy to evaluate how close a produced solution is to an optimal one for a given problem instance.
In this paper, we develop new data-dependent upper bounds for submodular maximization with a knapsack constraint. We theoretically prove that they dominate the optimal solution and empirically demonstrate their advantages in certifying how close to optimal a solution is through experiments with real-world datasets.
Blogger's Review: This paper introduces data-dependent upper bounds, providing a more precise evaluation tool for submodular maximization problems. It holds significant theoretical and practical value, particularly in complex machine learning tasks. The combination of theoretical proofs and experimental results makes this research highly relevant in the field of algorithm optimization.