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

2026-09-22

AI Summary

该研究解决了非单调次模最大化背包约束问题，通过线性查询复杂度的确定性算法，实现了1/4-ε近似解，提高了查询效率。

Abstract

Submodular maximization under a knapsack constraint (SMK) is a fundamental combinatorial optimization problem with broad applications across machine learning and data mining. Motivated by large-scale applications where query efficiency is paramount, we study non-monotone SMK and focus on deterministic algorithms with linear query complexity. Prior deterministic linear-query algorithms achieve at best a $1/5-\varepsilon$ approximation, falling short of the $1/4-\varepsilon$ ratio attainable by randomized algorithms. We close this gap by presenting a deterministic $(1/4-\varepsilon)$-approximation with $O(n\log^2(1/\varepsilon)/\varepsilon^2)$ queries. Our approach partitions the analysis based on the cost of the largest optimal element $r$: when the cost of $r$ is moderate, we refine the threshold-twin-greedy framework via residual-budget enumeration to tighten the analysis; when the cost of $r$ is large, we reduce the problem to bicriteria submodular maximization. As a secondary contribution, we obtain a $(1/2-\varepsilon, O(1/\varepsilon))$-bicriteria approximation with $O(n\log(1/\varepsilon)/\varepsilon^2)$ queries, improving over the previous $O(n^2/\varepsilon)$ query bound.

Problem

• Submodular Maximization
• Knapsack Constraint
• Deterministic Algorithms
• Linear Query Complexity

Innovation

• deterministic algorithm
• linear query complexity
• submodular maximization
• knapsack constraint
• bicriteria approximation