We investigate polynomial-time approximation schemes for the classic 0-1 knapsack problem. The previous algorithm by Deng, Jin, and Mao (SODA'23) has approximation factor $1 + \eps$ with running time $\widetilde{O}(n + \frac{1}{\eps^{2.2}})$. There is a lower Bound of $(n + \frac{1}{\eps})^{2-o(1)}$ conditioned on the hypothesis that $(\min, +)$ has no truly subquadratic algorithm. We close the gap by proposing an approximation scheme that runs in $\widetilde{O}(n + \frac{1}{\eps^2})$ time.

翻译：我们研究经典0-1背包问题的多项式时间近似方案。此前Deng、Jin和Mao（SODA'23）的算法在近似因子为$1 + \eps$时，运行时间为$\widetilde{O}(n + \frac{1}{\eps^{2.2}})$。基于$(\min, +)$不存在真正次二次算法的假设，存在一个下界$(n + \frac{1}{\eps})^{2-o(1)}$。我们通过提出一个运行时间为$\widetilde{O}(n + \frac{1}{\eps^2})$的近似方案来弥补这一差距。