Stronger Hardness for Submodular Maximization Subject to a Matroid Constraint

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.

13 pages. This version adds a remark about the relationship between this work and the results of arXiv:2610.00545