We show that the permanent of an $n\times n$ matrix of $\operatorname{poly}(n)$-bit integers and the number of Hamiltonian cycles of an $n$-vertex graph can both be computed in time $2^{n-\Omega(\sqrt{n})}$, improving an earlier algorithm of Bj\"orklund, Kaski, and Williams (Algorithmica 2019) that runs in time $2^{n - \Omega\left(\sqrt{n/\log \log n}\right)}$. A key tool of our approach is to design a data structure that supports fast "$r$-order evaluation" of permanent and Hamiltonian cycles, which cooperates with the new approach on multivariate multipoint evaluation by Bhargava, Ghosh, Guo, Kumar, and Umans (FOCS 2022).

翻译：我们证明，一个具有$\operatorname{poly}(n)$比特整数的$n\times n$矩阵的Permanent值，以及一个$n$顶点图的哈密顿环数量，均可在$2^{n-\Omega(\sqrt{n})}$时间内计算，这改进了Björklund、Kaski和Williams（Algorithmica 2019）的早期算法，该算法运行时间为$2^{n - \Omega\left(\sqrt{n/\log \log n}\right)}$。我们方法的一个关键工具是设计一种数据结构，支持永久值和哈密顿环的快速“$r$阶求值”，该结构与Bhargava、Ghosh、Guo、Kumar和Umans（FOCS 2022）关于多元多点求值的新方法相配合。