Deterministic Algorithm for Non-monotone Submodular Maximization under Matroid and Knapsack Constraints

2026-03-12

AI Summary

This work addresses the maximization of non-monotone submodular functions under both matroid and knapsack constraints. The authors propose a deterministic approximation algorithm based on an extended multilinear extension, which integrates deterministic continuous optimization with efficient discretization techniques. This approach achieves improved approximation ratios for both constraint types simultaneously while maintaining polynomial query complexity—the first such result in the deterministic setting. Specifically, the algorithm attains a (0.385 − ε)-approximation under matroid constraints and a (0.367 − ε)-approximation under knapsack constraints, surpassing the previous best deterministic guarantees of 0.367 and 0.25, respectively, and establishing the current state-of-the-art deterministic performance for these problems.

Technology Category

• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Submodular maximization constitutes a prominent research topic in combinatorial optimization and theoretical computer science, with extensive applications across diverse domains. While substantial advancements have been achieved in approximation algorithms for submodular maximization, the majority of algorithms yielding high approximation guarantees are randomized. In this work, we investigate deterministic approximation algorithms for maximizing non-monotone submodular functions subject to matroid and knapsack constraints. For the two distinct constraint settings, we propose novel deterministic algorithms grounded in an extended multilinear extension framework. Under matroid constraints, our algorithm achieves an approximation ratio of $(0.385 - ε)$, whereas for knapsack constraints, the proposed algorithm attains an approximation ratio of $(0.367 -ε)$. Both algorithms run in $\mathrm{poly}(n)$ query complexity, where $n$ is the size of the ground set, and improve upon the state-of-the-art deterministic approximation ratios of $(0.367 - ε)$ for matroid constraints and $0.25$ for knapsack constraints.

Problem

• submodular maximization
• non-monotone
• matroid constraints
• knapsack constraints
• deterministic algorithm

Innovation

• deterministic algorithm
• non-monotone submodular maximization
• matroid constraint
• knapsack constraint
• multilinear extension