In this work, we study the problem of finding the maximum value of a non-negative submodular function subject to a limit on the number of items selected, a ubiquitous problem that appears in many applications, such as data summarization and nonlinear regression. We provide the first deterministic, linear-time approximation algorithms for this problem that do not assume the objective is monotone. We present three deterministic, linear-time algorithms: a single-pass streaming algorithm with a ratio of $23.313 + \epsilon$, which is the first linear-time streaming algorithm; a simpler deterministic linear-time algorithm with a ratio of $11.657$; and a $(4 + O(\epsilon ))$-approximation algorithm. Finally, we present a deterministic algorithm that obtains ratio of $e + \epsilon$ in $O_{\epsilon}(n \log(n))$ time, close to the best known expected ratio of $e - 0.121$ in polynomial time.

翻译：本文研究了在项目数量限制下最大化非负子模函数值的问题，这是出现在数据摘要和非线性回归等众多应用中的普遍问题。我们首次提出了不假设目标函数为单调函数的确定性线性时间近似算法。我们给出了三种确定性线性时间算法：比率为$23.313 + \epsilon$的单遍流算法，这是首个线性时间流算法；比率为$11.657$的更简洁的确定性线性时间算法；以及一个$(4 + O(\epsilon ))$-近似算法。最后，我们提出了一种确定性算法，在$O_{\epsilon}(n \log(n))$时间内达到$e + \epsilon$的比率，接近多项式时间内已知最优期望比率$e - 0.121$。