  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.
