We consider the problem of maximizing a non-negative monotone submodular function subject to a knapsack constraint, which is also known as the Budgeted Submodular Maximization (BSM) problem. Sviridenko (2004) showed that by guessing 3 appropriate elements of an optimal solution, and then executing a greedy algorithm, one can obtain the optimal approximation ratio of $\alpha =1-1/e\approx 0.632$ for BSM. However, the need to guess (by enumeration) 3 elements makes the algorithm of Sviridenko impractical as it leads to a time complexity of $O(n^5)$ (which can be slightly improved using the thresholding technique of Badanidiyuru & Vondrak (2014), but only to roughly $O(n^4)$). Our main results in this paper show that fewer guesses suffice. Specifically, by making only 2 guesses, we get the same optimal approximation ratio of $\alpha$ with an improved time complexity of roughly $O(n^3)$. Furthermore, by making only a single guess, we get an almost as good approximation ratio of $0.6174>0.9767\alpha$ in roughly $O(n^2)$ time. Prior to our work, the only algorithms that were known to obtain an approximation ratio close to $\alpha$ for BSM were the algorithm of Sviridenko and an algorithm of Ene & Nguyen (2019) that achieves $(\alpha-\epsilon)$-approximation. However, the algorithm of Ene & Nguyen requires ${(1/\epsilon)}^{O(1/\epsilon^4)}n\log^2 n$ time, and hence, is of theoretical interest only as ${(1/\epsilon)}^{O(1/\epsilon^4)}$ is huge even for moderate values of $\epsilon$. In contrast, all the algorithms we analyze are simple and parallelizable, which makes them good candidates for practical use. Recently, Tang et al. (2020) studied a simple greedy algorithm that already has a long research history, and proved that its approximation ratio is at least 0.405. We improve over this result, and show that the approximation ratio of this algorithm is within the range [0.427, 0.462].

翻译：我们考虑的是将非负数单调子模式功能最大化的问题。 然而, 需要( 通过罗列) 3 个元素使Sviridenko的算法变得不切实际, 因为它导致一个时间复杂性$O( n%5) 美元( 通过Badanidiyur & Vondrak(2014)的临界技术可以稍有改善, 但只有大约 $O( n=4) 。 我们本文的主要结果显示, $alpha=1/ e\ appro0. 632美元的最佳近似比率。 然而, 只需要猜测( 通过罗列) 3个元素使Sviridenko的算法变得不切实际, 因为它导致一个时间复杂性 $( n%5 美元 ) (可以通过 Badrididiaryruru和 Vondrakrak(2014)的临界值略改进 ) 。 本文的主要结果显示, 仅仅通过两次猜测, 美元( n=3) 美元( 美元) 和 美元( 美元) (n&xial 美元) 美元(national) yal) yal__xal=974\) 的算算算算算算算。