Stronger Hardness for Submodular Maximization Subject to a Matroid Constraint

2026-10-05

AI Summary

This study investigates the hardness bottleneck of approximation algorithms for submodular maximization under matroid constraints. Operating within the value oracle model and leveraging combinatorial optimization theory, we construct rigorous computational complexity reductions to demonstrate that the approximation ratio of any polynomial-time algorithm cannot exceed 8/17 (approximately 0.471), even under simplified constraints such as cardinality or partition matroids. This result achieves the first breakthrough in a long-stagnant theoretical lower bound in fifteen years, substantially narrowing the gap between algorithmic performance and theoretical limits. Furthermore, it establishes a more precise theoretical benchmark for the current best 0.401-approximation algorithm, thereby advancing the broader field of submodular optimization.

Abstract

Maximizing a submodular function subject to a matroid constraint is a cornerstone problem in combinatorial optimization. The state-of-the-art algorithm for this problem obtains $0.401$-approximation~\cite{buchbinder2024constrained}, and the state-of-the-art hardness result shows that no polynomial time algorithm can obtain better than $0.478$-approximation for this problem~\cite{oveisgharan2011submodular}. In this work, we present the first improvement in $15$ years for the hardness result, showing that no polynomial time algorithm in the value-oracle model can obtain better than $8/17 \approx 0.471$-approximation, even for the special case of a cardinality or (simplified) partition matroid constraint.

Problem

• submodular maximization
• matroid constraint
• hardness of approximation
• combinatorial optimization
• value-oracle model

Innovation

• Submodular Maximization
• Matroid Constraint
• Hardness of Approximation
• Value-Oracle Model
• Combinatorial Optimization