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

TL;DR

This work advances submodular maximization under complex constraints by breaking the greedy barrier for matroid k-parity constraints with a hybrid greedy-local-search algorithm. The core technique partitions weight classes and introduces auxiliary weights to enable a robust charging argument that yields a multiplicative improvement over the classical greedy bound, achieving a ratio near 0.819k for monotone submodular objectives (and 0.694k for linear objectives) while remaining polynomial-time and independent of k. It also extends to non-monotone submodular objectives via a non-monotone extension that preserves a similar multiplicative scale, and the framework generalizes to matroid k-intersection and k-parity constraints. The results offer a principled, scalable approach for large-scale combinatorial optimization where the constraint structure is highly expressive, with potential impact on scheduling, matching, and resource allocation problems under complex feasibility criteria.

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.

Figures (1)

• Figure 1: The four families of constraints discussed in Section \ref{['sec:introduction']}. An arrow from family A of constraints to family B indicates that family B generalizes family A. Below each constraint family, we list the state-of-the-art approximation ratios for maximizing cardinality (C), linear (L), monotone submodular (MS) and/or non-monotone submodular (NS) objectives subject to constraints of this family. New results of this paper are marked with a grey background (and the results they improve over appear below them in parentheses). To avoid repetition, the approximation ratios for $k$-Dimensional Matching constraints are omitted as they are identical to the ratios for the more general $k$-Set Packing constraints.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• proof
• Definition 3.1: $(\theta, \varepsilon)$-improvement
• proof
• proof
• Definition 4.1
• proof