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.

翻译：本文采用阈值递减算法解决拟阵约束下的$k$-子模函数最大化问题，与贪心算法相比，该算法在近似比损失较小的同时显著降低了查询复杂度。针对单调$k$-子模函数最大化，我们给出了$(\frac{1}{2} - \epsilon)$-近似算法；针对非单调情形，给出了$(\frac{1}{3} - \epsilon)$-近似算法，其复杂度为$O(\frac{n(k\cdot EO + IO)}{\epsilon} \log \frac{r}{\epsilon})$，其中$r$表示拟阵的秩，$IO$和$EO$分别表示判断子集是否为独立集的预言机及计算函数值$f$的预言机调用次数。由于总量约束可视为特殊拟阵（即均匀拟阵），我们进而给出了总量约束下$k$-子模函数最大化的快速算法作为推论。