Submodular Maximization over a Matroid $k$-Intersection: Multiplicative Improvement over Greedy

2026-02-09

AI Summary

This work addresses the problem of maximizing a non-negative submodular function subject to the intersection of $k$ matroid constraints. The authors propose a hybrid greedy local search algorithm that, for the first time in the general $k$-constraint setting, achieves a multiplicative improvement over the classical greedy approximation ratio while maintaining a runtime dependent only on the size of the ground set and independent of $k$. Specifically, for monotone functions, the algorithm attains an approximation ratio of $0.819k + O(\sqrt{k})$, and for non-monotone functions, $0.819k + O(k^{2/3})$, both improving upon the best-known results. Furthermore, the proposed framework naturally extends to non-monotone objectives and more general matroid $k$-parity constraints.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• 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
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We study the problem of maximizing a non-negative monotone submodular objective $f$ subject to the intersection of $k$ arbitrary matroid constraints. The natural greedy algorithm guarantees $(k+1)$-approximation for this problem, and the state-of-the-art algorithm only improves this approximation ratio to $k$. We give a $\frac{2k\ln2}{1+\ln2}+O(\sqrt{k})<0.819k+O(\sqrt{k})$ approximation for this problem. Our result is the first multiplicative improvement over the approximation ratio of the greedy algorithm for general $k$. We further show that our algorithm can be used to obtain roughly the same approximation ratio also for the more general problem in which the objective is not guaranteed to be monotone (the sublinear term in the approximation ratio becomes $O(k^{2/3})$ rather than $O(\sqrt{k})$ in this case). All of our results hold also when the $k$-matroid intersection constraint is replaced with a more general matroid $k$-parity constraint. Furthermore, unlike the case in many of the previous works, our algorithms run in time that is independent of $k$ and polynomial in the size of the ground set. Our algorithms are based on a hybrid greedy local search approach recently introduced by Singer and Thiery (STOC 2025) for the weighted matroid $k$-intersection problem, which is a special case of the problem we consider. Leveraging their approach in the submodular setting requires several non-trivial insights and algorithmic modifications since the marginals of a submodular function $f$, which correspond to the weights in the weighted case, are not independent of the algorithm's internal randomness. In the special weighted case studied by Singer and Thiery, our algorithms reduce to a variant of their algorithm with an improved approximation ratio of $k\ln2+1-\ln2<0.694k+0.307$, compared to an approximation ratio of $\frac{k+1}{2\ln2}\approx0.722k+0.722$ guaranteed by Singer and Thiery.

Problem

• submodular maximization
• matroid intersection
• approximation algorithm
• monotone submodular function
• matroid k-parity

Innovation

• submodular maximization
• matroid intersection
• approximation algorithm
• greedy local search
• multiplicative improvement