We present a pseudopolynomial-time algorithm for the Knapsack problem that has running time $\widetilde{O}(n + t\sqrt{p_{\max}})$, where $n$ is the number of items, $t$ is the knapsack capacity, and $p_{\max}$ is the maximum item profit. This improves over the $\widetilde{O}(n + t \, p_{\max})$-time algorithm based on the convolution and prediction technique by Bateni et al.~(STOC 2018). Moreover, we give some evidence, based on a strengthening of the Min-Plus Convolution Hypothesis, that our running time might be optimal. Our algorithm uses two new technical tools, which might be of independent interest. First, we generalize the $\widetilde{O}(n^{1.5})$-time algorithm for bounded monotone min-plus convolution by Chi et al.~(STOC 2022) to the \emph{rectangular} case where the range of entries can be different from the sequence length. Second, we give a reduction from general knapsack instances to \emph{balanced} instances, where all items have nearly the same profit-to-weight ratio, up to a constant factor. Using these techniques, we can also obtain algorithms that run in time $\widetilde{O}(n + OPT\sqrt{w_{\max}})$, $\widetilde{O}(n + (nw_{\max}p_{\max})^{1/3}t^{2/3})$, and $\widetilde{O}(n + (nw_{\max}p_{\max})^{1/3} OPT^{2/3})$, where $OPT$ is the optimal total profit and $w_{\max}$ is the maximum item weight.

翻译：我们提出了一种面向背包问题的伪多项式时间算法，其运行时间为$\widetilde{O}(n + t\sqrt{p_{\max}})$，其中$n$为物品数量，$t$为背包容量，$p_{\max}$为最大物品利润。该结果改进了Bateni等人（STOC 2018）基于卷积与预测技术提出的$\widetilde{O}(n + t \, p_{\max})$时间复杂度算法。此外，基于最小-加卷积假设的强化版本，我们给出了该运行时间可能已达到最优的若干证据。我们的算法采用了两种可能具有独立价值的新技术：首先，我们将Chi等人（STOC 2022）针对有界单调最小-加卷积的$\widetilde{O}(n^{1.5})$时间复杂度算法推广至\emph{矩形}情形，其中数值取值范围可与序列长度不同；其次，我们给出了一种从一般背包实例到\emph{平衡}实例的归约方法，在平衡实例中所有物品的利润-重量比在常数因子范围内近似相等。利用这些技术，我们还可获得运行时间分别为$\widetilde{O}(n + OPT\sqrt{w_{\max}})$、$\widetilde{O}(n + (nw_{\max}p_{\max})^{1/3}t^{2/3})$和$\widetilde{O}(n + (nw_{\max}p_{\max})^{1/3} OPT^{2/3})$的算法，其中$OPT$为最优总利润，$w_{\max}$为最大物品重量。