Data-dependent Evaluations for Budgeted Submodular Maximization

2026-07-06

AI Summary

This work addresses the challenge of evaluating solution quality for submodular maximization under a knapsack constraint, where existing algorithms offer only conservative worst-case approximation guarantees that poorly reflect practical performance. The paper introduces, for the first time, a data-dependent upper bound that is theoretically guaranteed to be strictly tighter than the trivial bound on the optimal value. By integrating submodular optimization theory with data-driven techniques, the authors develop a novel upper-bound estimation framework tailored to knapsack-constrained settings. Extensive experiments on multiple real-world datasets demonstrate that the proposed bound provides a significantly tighter approximation of the true optimum, thereby substantially enhancing the certifiable quality of obtained solutions and markedly outperforming conventional worst-case bounds.

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.

Problem

• submodular maximization
• knapsack constraint
• data-dependent evaluation
• approximation guarantee
• NP-hardness

Innovation

• data-dependent bounds
• submodular maximization
• knapsack constraint
• instance-specific evaluation
• approximation guarantee