Given a natural number $k\ge 2$, we consider the $k$-submodular cover problem ($k$-SC). The objective is to find a minimum cost subset of a ground set $\mathcal{X}$ subject to the value of a $k$-submodular utility function being at least a certain predetermined value $\tau$. For this problem, we design a bicriteria algorithm with a cost at most $O(1/\epsilon)$ times the optimal value, while the utility is at least $(1-\epsilon)\tau/r$, where $r$ depends on the monotonicity of $g$.

翻译：给定自然数$k\ge 2$，我们考虑$k$-子模覆盖问题（$k$-SC）。该目标是在基础集$\mathcal{X}$上寻找满足$k$-子模效用函数值至少达到预定阈值$\tau$的最小成本子集。针对此问题，我们设计了一个双准则算法，其成本不超过最优值的$O(1/\epsilon)$倍，同时效用至少达到$(1-\epsilon)\tau/r$，其中$r$取决于函数$g$的单调性。