We show that the hafnian of a symmetric $2n\times 2n$ matrix of $\operatorname{poly}(n)$-bit integers (which counts the number of perfect matchings of a $2n$-vertex graph) and the number of Hamiltonian cycles of an $n$-vertex directed graph can be computed in time $2^{n-Ω(\sqrt{n})}$, improving and generalizing an earlier algorithm of Björklund, Kaski, and Williams (Algorithmica 2019) that runs in time $2^{n - Ω\left(\sqrt{n/\log \log n}\right)}$. A key tool of our approach is the design of a data structure that supports fast evaluation of high-order derivatives of hafnian and Hamiltonian cycles, which integrates with the new approach on multivariate multipoint evaluation by Bhargava, Ghosh, Guo, Kumar, and Umans (FOCS 2022, JACM 2024).

翻译：我们证明：一个对称 $2n\times 2n$ 矩阵（其元素为 $\operatorname{poly}(n)$ 比特整数）的缩并数（用于计算 $2n$ 顶点图中完美匹配的数量）以及 $n$ 顶点有向图中哈密顿圈的数量，可在 $2^{n-Ω(\sqrt{n})}$ 时间内计算。这改进了 Björklund、Kaski 与 Williams（Algorithmica 2019）先前提出的 $2^{n - Ω\left(\sqrt{n/\log \log n}\right)}$ 时间算法，并对其进行了推广。我们方法的一个关键工具是设计一种数据结构，支持对缩并数与哈密顿圈高阶导数的快速求值，该数据结构与 Bhargava、Ghosh、Guo、Kumar 和 Umans（FOCS 2022, JACM 2024）提出的多变量多点求值新方法相结合。