# On maximizing a monotone k-submodular function subject to a matroid constraint

arXiv: https://arxiv.org/abs/1607.07957 · HTML: https://mlsanity.com/paper/1607.07957/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jul 2016.

## Built on by (most cited first, 6 of 6)

- [Stochastic k-Submodular Bandits with Full Bandit Feedback](https://mlsanity.com/paper/2412.10682/) (2024)
- [Fast algorithms for k-submodular maximization subject to a matroid constraint](https://mlsanity.com/paper/2307.13996/) (2023)
- [Robust Approximation Algorithms for Non-monotone k-Submodular Maximization under a Knapsack Constraint](https://mlsanity.com/paper/2309.12025/) (2023)
- [k-Submodular Interdiction Problems under Distributional Risk-Receptiveness and Robustness: Application to Machine Learning](https://mlsanity.com/paper/2406.13023/) (2024)
- [Fairness in Monotone k-submodular Maximization: Algorithms and Applications](https://mlsanity.com/paper/2411.05318/) (2024)
- [Fast Stochastic Greedy Algorithm for k-Submodular Cover Problem](https://mlsanity.com/paper/2511.00869/) (2025)

## Abstract (arXiv)

A k-submodular function is an extension of a submodular function in that its input is given by k disjoint subsets instead of a single subset. For unconstrained nonnegative k-submodular maximization, Ward and vZivn'y proposed a constant-factor approximation algorithm, which was improved by the recent work of Iwata, Tanigawa and Yoshida presenting a 1/2-approximation algorithm. Iwata et al. also provided a k/(2k-1)-approximation algorithm for monotone k-submodular maximization and proved that its approximation ratio is asymptotically tight. More recently, Ohsaka and Yoshida proposed constant-factor algorithms for monotone k-submodular maximization with several size constraints. However, while submodular maximization with various constraints has been extensively studied, no approximation algorithm has been developed for constrained k-submodular maximization, except for the case of size constraints. In this paper, we prove that a greedy algorithm outputs a 1/2-approximate solution for monotone k-submodular maximization with a matroid constraint. The algorithm runs in O(M|E|(MO + kEO)) time, where M is the size of a maximal optimal solution, |E| is the size of the ground set, and MO, EO represent the time for the membership oracle of the matroid and the evaluation oracle of the k-submodular function, respectively.