  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 \v{Z}ivn\'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 $O(M|E|(\text{MO} + k\text{EO}))$
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.
