The problem of non-monotone $k$-submodular maximization under a knapsack constraint ($\kSMK$) over the ground set size $n$ has been raised in many applications in machine learning, such as data summarization, information propagation, etc. However, existing algorithms for the problem are facing questioning of how to overcome the non-monotone case and how to fast return a good solution in case of the big size of data. This paper introduces two deterministic approximation algorithms for the problem that competitively improve the query complexity of existing algorithms. Our first algorithm, $\LAA$, returns an approximation ratio of $1/19$ within $O(nk)$ query complexity. The second one, $\RLA$, improves the approximation ratio to $1/5-\epsilon$ in $O(nk)$ queries, where $\epsilon$ is an input parameter. Our algorithms are the first ones that provide constant approximation ratios within only $O(nk)$ query complexity for the non-monotone objective. They, therefore, need fewer the number of queries than state-of-the-the-art ones by a factor of $\Omega(\log n)$. Besides the theoretical analysis, we have evaluated our proposed ones with several experiments in some instances: Influence Maximization and Sensor Placement for the problem. The results confirm that our algorithms ensure theoretical quality as the cutting-edge techniques and significantly reduce the number of queries.

翻译：在背包约束下的非单调$k$-子模最大化问题（$\kSMK$），其定义于基集规模$n$上，已在机器学习领域的诸多应用中涌现，例如数据摘要、信息传播等。然而，现有算法面临如何应对非单调情形以及如何在数据规模庞大时快速返回优质解的挑战。本文针对该问题提出了两种确定性近似算法，在查询复杂度上较现有算法实现了竞争性改进。第一种算法$\LAA$在$O(nk)$查询复杂度内达到$1/19$的近似比；第二种算法$\RLA$将近似比提升至$1/5-\epsilon$（其中$\epsilon$为输入参数），仍保持$O(nk)$查询复杂度。本文算法是首个在非单调目标下仅需$O(nk)$查询复杂度便能提供常数近似比的方法，因此其查询次数较现有最优方法减少$\Omega(\log n)$倍。除理论分析外，我们针对影响力最大化和传感器布设两类实例对所提算法进行了多组实验评估。结果表明，该算法在保证与前沿技术相当的理论性能的同时，显著降低了查询次数。