In this paper, we apply a Threshold-Decreasing Algorithm to maximize $k$-submodular functions under a matroid constraint, which reduces the query complexity of the algorithm compared to the greedy algorithm with little loss in approximation ratio. We give a $(\frac{1}{2} - \epsilon)$-approximation algorithm for monotone $k$-submodular function maximization, and a $(\frac{1}{3} - \epsilon)$-approximation algorithm for non-monotone case, with complexity $O(\frac{n(k\cdot EO + IO)}{\epsilon} \log \frac{r}{\epsilon})$, where $r$ denotes the rank of the matroid, and $IO, EO$ denote the number of oracles to evaluate whether a subset is an independent set and to compute the function value of $f$, respectively. Since the constraint of total size can be looked as a special matroid, called uniform matroid, then we present the fast algorithm for maximizing $k$-submodular functions subject to a total size constraint as corollaries. corollaries.

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