Approximating Partition in Near-Linear Time

We propose an $\tilde{O}(n+1\mathrm{/}\text{eps})$-time FPTAS (Fully Polynomial-Time Approximation Scheme) for the classical Partition problem. This is the best possible (up to a polylogarithmic 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 $\tilde{O}(n+1\mathrm{/}{\text{eps}}^{5\mathrm{/}4})$ 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.