Robust Approximation Algorithms for Non-monotone $k$-Submodular Maximization under a Knapsack Constraint

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.