A Nearly Quadratic-Time FPTAS for Knapsack

Abstract

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