The Bounded Knapsack problem is one of the most fundamental NP-complete problems at the intersection of computer science, optimization, and operations research. A recent line of research worked towards understanding the complexity of pseudopolynomial-time algorithms for Bounded Knapsack parameterized by the maximum item weight $w_{\mathrm{max}}$ and the number of items $n$. A conditional lower bound rules out that Bounded Knapsack can be solved in time $O((n+w_{\mathrm{max}})^{2-\delta})$ for any $\delta > 0$ [Cygan, Mucha, Wegrzycki, Wlodarczyk'17, K\"unnemann, Paturi, Schneider'17]. This raised the question whether Bounded Knapsack can be solved in time $\tilde O((n+w_{\mathrm{max}})^2)$. The quest of resolving this question lead to algorithms that run in time $\tilde O(n^3 w_{\mathrm{max}}^2)$ [Tamir'09], $\tilde O(n^2 w_{\mathrm{max}}^2)$ and $\tilde O(n w_{\mathrm{max}}^3)$ [Bateni, Hajiaghayi, Seddighin, Stein'18], $O(n^2 w_{\mathrm{max}}^2)$ and $\tilde O(n w_{\mathrm{max}}^2)$ [Eisenbrand and Weismantel'18], $O(n + w_{\mathrm{max}}^3)$ [Polak, Rohwedder, Wegrzycki'21], and very recently $\tilde O(n + w_{\mathrm{max}}^{12/5})$ [Chen, Lian, Mao, Zhang'23]. In this paper we resolve this question by designing an algorithm for Bounded Knapsack with running time $\tilde O(n + w_{\mathrm{max}}^2)$, which is conditionally near-optimal.

翻译：有界背包问题是计算机科学、优化与运筹学交叉领域中最基础的NP完全问题之一。近期一系列研究致力于理解以最大物品重量$w_{\mathrm{max}}$和物品数量$n$为参数的有界背包问题的伪多项式时间算法复杂度。条件性下界表明，对于任意$\delta > 0$，有界背包问题无法在$O((n+w_{\mathrm{max}})^{2-\delta})$时间内求解 [Cygan, Mucha, Wegrzycki, Wlodarczyk'17, Künnemann, Paturi, Schneider'17]。这引发了有界背包问题能否在$\tilde O((n+w_{\mathrm{max}})^2)$时间内求解的疑问。对该问题的探索催生了一系列算法，其运行时间分别为$\tilde O(n^3 w_{\mathrm{max}}^2)$ [Tamir'09]、$\tilde O(n^2 w_{\mathrm{max}}^2)$和$\tilde O(n w_{\mathrm{max}}^3)$ [Bateni, Hajiaghayi, Seddighin, Stein'18]、$O(n^2 w_{\mathrm{max}}^2)$和$\tilde O(n w_{\mathrm{max}}^2)$ [Eisenbrand and Weismantel'18]、$O(n + w_{\mathrm{max}}^3)$ [Polak, Rohwedder, Wegrzycki'21]，以及近期提出的$\tilde O(n + w_{\mathrm{max}}^{12/5})$ [Chen, Lian, Mao, Zhang'23]。本文通过设计运行时间为$\tilde O(n + w_{\mathrm{max}}^2)$的有界背包问题算法解决了该疑问，该时间复杂度在条件性意义下近乎最优。