On an orientable surface $S$, consider a collection $\Gamma$ of closed curves. The (geometric) intersection number $i_S(\Gamma)$ is the minimum number of self-intersections that a collection $\Gamma'$ can have, where $\Gamma'$ results from a continuous deformation (homotopy) of $\Gamma$. We provide algorithms that compute $i_S(\Gamma)$ and such a $\Gamma'$, assuming that $\Gamma$ is given by a collection of closed walks of length $n$ in a graph $M$ cellularly embedded on $S$, in $O(n \log n)$ time when $M$ and $S$ are fixed. The state of the art is a paper of Despr\'e and Lazarus [SoCG 2017, J. ACM 2019], who compute $i_S(\Gamma)$ in $O(n^2)$ time, and $\Gamma'$ in $O(n^4)$ time if $\Gamma$ is a single closed curve. Our result is more general since we can put an arbitrary number of closed curves in minimal position. Also, our algorithms are quasi-linear in $n$ instead of quadratic and quartic, and our proofs are simpler and shorter. We use techniques from two-dimensional topology and from the theory of hyperbolic surfaces. Most notably, we prove a new property of the reducing triangulations introduced by Colin de Verdi\`ere, Despr\'e, and Dubois [SODA 2024], reducing our problem to the case of surfaces with boundary. As a key subroutine, we rely on an algorithm of Fulek and T\'oth [JCO 2020].

翻译：在可定向曲面$S$上，考虑一个封闭曲线集合$\Gamma$。其（几何）交点数$i_S(\Gamma)$是指通过连续变形（同伦）将$\Gamma$变为$\Gamma'$后，$\Gamma'$所能达到的最小自交点数。我们提出算法来计算$i_S(\Gamma)$及相应的$\Gamma'$，假设$\Gamma$由长度为$n$的闭迹集合给出，这些闭迹位于图$M$中（$M$细胞嵌入在$S$上），当$M$和$S$固定时，算法时间复杂度为$O(n \log n)$。当前最优算法是Despré与Lazarus的论文[SoCG 2017, J. ACM 2019]，其中计算$i_S(\Gamma)$需$O(n^2)$时间，若$\Gamma$为单条封闭曲线，计算$\Gamma'$需$O(n^4)$时间。我们的结果更具一般性，因为可将任意数量的封闭曲线置于最小位置。此外，我们的算法在$n$上为准线性而非二次或四次，且证明更简洁简短。我们使用了二维拓扑学与双曲曲面理论中的技术。尤为重要的是，我们证明了Colin de Verdière、Despré和Dubois [SODA 2024]引入的约化三角剖分的一个新性质，从而将问题简化为带边曲面的情形。作为关键子程序，我们依赖于Fulek与Tóth [JCO 2020]的算法。