We propose an $\widetilde{O}(n + 1/\varepsilon)$-time FPTAS (Fully Polynomial-Time Approximation Scheme) for the classical Partition problem. This is the best possible (up to a logarithmic factor) assuming SETH (Strong Exponential Time Hypothesis) [Abboud, Bringmann, Hermelin, and Shabtay'22]. Prior to our work, the best known FPTAS for Partition runs in $\widetilde{O}(n + 1/\varepsilon^{5/4})$ time [Deng, Jin and Mao'23, Wu and Chen'22]. Our result is obtained by solving a more general problem of weakly approximating Subset Sum.

翻译：我们提出了一个经典划分问题的$\widetilde{O}(n + 1/\varepsilon)$时间FPTAS（完全多项式时间近似方案）。在SETH（强指数时间假设）下，这是可能达到的最优时间复杂度（相差不超过对数因子）[Abboud, Bringmann, Hermelin, and Shabtay'22]。此前，划分问题最快的已知FPTAS运行时间为$\widetilde{O}(n + 1/\varepsilon^{5/4})$[Deng, Jin and Mao'23, Wu and Chen'22]。我们的结果是通过解决一个更一般的问题——子集和弱近似——而获得的。