In this paper, we study the problem of maximizing $k$-submodular functions subject to a knapsack constraint. For monotone objective functions, we present a $\frac{1}{2}(1-e^{-2})\approx 0.432$ greedy approximation algorithm. For the non-monotone case, we are the first to consider the knapsack problem and provide a greedy-type combinatorial algorithm with approximation ratio $\frac{1}{3}(1-e^{-3})\approx 0.317$.

翻译：本文研究受背包约束的$k$-子模函数最大化问题。针对单调目标函数，我们提出一个$\frac{1}{2}(1-e^{-2})\approx 0.432$的贪心近似算法。对于非单调情形，我们首次考虑该背包问题，并给出一个近似比为$\frac{1}{3}(1-e^{-3})\approx 0.317$的贪心型组合算法。