Approximating Partition in Near-Linear Time

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.