In this paper, we obtain a number of new simple pseudo-polynomial time algorithms on the well-known knapsack problem, focusing on the running time dependency on the number of items $n$, the maximum item weight $w_\mathrm{max}$, and the maximum item profit $p_\mathrm{max}$. Our results include: - An $\widetilde{O}(n^{3/2}\cdot \min\{w_\mathrm{max},p_\mathrm{max}\})$-time randomized algorithm for 0-1 knapsack, improving the previous $\widetilde{O}(\min\{n w_\mathrm{max} p_\mathrm{max}^{2/3},n p_\mathrm{max} w_\mathrm{max}^{2/3}\})$ [Bringmann and Cassis, ESA'23] for the small $n$ case. - An $\widetilde{O}(n+\min\{w_\mathrm{max},p_\mathrm{max}\}^{5/2})$-time randomized algorithm for bounded knapsack, improving the previous $O(n+\min\{w_\mathrm{max}^3,p_\mathrm{max}^3\})$ [Polak, Rohwedder and Wegrzyck, ICALP'21].

翻译：本文针对经典背包问题提出了一系列新的简单伪多项式时间算法，重点研究运行时间与物品数量$n$、最大物品重量$w_\mathrm{max}$及最大物品利润$p_\mathrm{max}$的依赖关系。主要成果包括： - 针对0-1背包问题的$\widetilde{O}(n^{3/2}\cdot \min\{w_\mathrm{max},p_\mathrm{max}\})$时间随机算法，在$n$较小的情况下改进了之前$\widetilde{O}(\min\{n w_\mathrm{max} p_\mathrm{max}^{2/3},n p_\mathrm{max} w_\mathrm{max}^{2/3}\})$的结果[Bringmann和Cassis, ESA'23]。 - 针对有界背包问题的$\widetilde{O}(n+\min\{w_\mathrm{max},p_\mathrm{max}\}^{5/2})$时间随机算法，改进了之前$O(n+\min\{w_\mathrm{max}^3,p_\mathrm{max}^3\})$的结果[Polak、Rohwedder和Węgrzyck, ICALP'21]。