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：我们研究了经典0-1背包问题的多项式时间近似方案。邓、金和毛的先前算法（SODA'23）具有近似因子$1 + \eps$，运行时间为$\widetilde{O}(n + \frac{1}{\eps^{2.2}})$。在假设$(\min, +)$不存在真正次二次算法的条件下，存在下界$(n + \frac{1}{\eps})^{2-o(1)}$。我们通过提出一个运行时间为$\widetilde{O}(n + \frac{1}{\eps^2})$的近似方案来弥合这一差距。