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

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 Živný 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 $O(M|E|(\text{MO} + k\text{EO}))$ time, where $M$ is the size of a maximal optimal solution, $|E|$ is the size of the ground set, and $\text{MO}, \text{EO}$ represent the time for the membership oracle of the matroid and the evaluation oracle of the $k$-submodular function, respectively.

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