This work, for the first time, introduces two constant factor approximation algorithms with linear query complexity for non-monotone submodular maximization over a ground set of size $n$ subject to a knapsack constraint, $\mathsf{DLA}$ and $\mathsf{RLA}$. $\mathsf{DLA}$ is a deterministic algorithm that provides an approximation factor of $6+\epsilon$ while $\mathsf{RLA}$ is a randomized algorithm with an approximation factor of $4+\epsilon$. Both run in $O(n \log(1/\epsilon)/\epsilon)$ query complexity. The key idea to obtain a constant approximation ratio with linear query lies in: (1) dividing the ground set into two appropriate subsets to find the near-optimal solution over these subsets with linear queries, and (2) combining a threshold greedy with properties of two disjoint sets or a random selection process to improve solution quality. In addition to the theoretical analysis, we have evaluated our proposed solutions with three applications: Revenue Maximization, Image Summarization, and Maximum Weighted Cut, showing that our algorithms not only return comparative results to state-of-the-art algorithms but also require significantly fewer queries.

翻译：本文首次针对大小为$n$的基集上的非单调子模最大化问题，在背包约束下引入两种具有线性查询复杂度的常数因子逼近算法：$\mathsf{DLA}$和$\mathsf{RLA}$。$\mathsf{DLA}$是一种确定性算法，提供$6+\epsilon$的逼近因子；而$\mathsf{RLA}$是一种随机算法，提供$4+\epsilon$的逼近因子。两种算法的查询复杂度均为$O(n \log(1/\epsilon)/\epsilon)$。实现线性查询下常数逼近比的关键思想在于：（1）将基集划分为两个合适的子集，通过线性查询在这些子集上寻找近似最优解；（2）将阈值贪心方法与两个不相交集合的性质或随机选择过程相结合，以提升解的质量。除理论分析外，我们还用三个应用场景评估了所提出的方案：收益最大化、图像摘要和最大加权割。结果表明，我们的算法不仅返回的结果与最先进算法相当，而且所需的查询数量显著更少。