The geometric intersection number of a curve on a surface is the minimal number of self-intersections of any homotopic curve, i.e., of any curve obtained by continuous deformation. Given a curve c represented by a closed walk of length at most ℓ on a combinatorial surface of complexity n , we describe simple algorithms to (1) compute the geometric intersection number of c in O ( n + ℓ 2 ) time, (2) construct a curve homotopic to c that realizes this geometric intersection number in O ( n +ℓ 4 ) time, and (3) decide if the geometric intersection number of c is zero, i.e., if c is homotopic to a simple curve, in O ( n +ℓ log ℓ) time. The algorithms for (2) and (3) are restricted to orientable surfaces, but the algorithm for (1) is also valid on non-orientable surfaces. To our knowledge, no exact complexity analysis had yet appeared on those problems. An optimistic analysis of the complexity of the published algorithms for problems (1) and (3) gives at best a O ( n + g 2 ℓ 2 ) time complexity on a genus g surface without boundary. No polynomial time algorithm was known for problem (2) for surfaces without boundary. Interestingly, our solution to problem (3) provides a quasi-linear algorithm to a problem raised by Poincaré more than a century ago. Finally, we note that our algorithm for problem (1) extends to computing the geometric intersection number of two curves of length at most ℓ in O ( n + ℓ 2 ) time.

中文翻译：

---

计算曲线的几何交点数

曲面上曲线的几何交点数是任意同伦曲线，即连续变形得到的任意曲线的最小自交点数。给定一条曲线C由在复杂组合表面上长度最多为 ℓ 的封闭游走表示n，我们描述了简单的算法来（1）计算几何交集数C在○(n+ ℓ2) 时间，(2) 构造一条同伦曲线C实现这个几何交点数○(n+ℓ4) 时间，和 (3) 确定几何交点数C为零，即，如果C与简单曲线同伦，在○(n+ℓ log ℓ) 时间。(2) 和 (3) 的算法仅限于可定向的表面，但 (1) 的算法也适用于不可定向的表面。据我们所知，尚未对这些问题进行精确的复杂性分析。对问题 (1) 和 (3) 的已发布算法的复杂性进行乐观分析，最多只能给出一个○(n+G 2ℓ2) 一个属的时间复杂度G无边界的表面。对于没有边界的表面，问题 (2) 没有多项式时间算法。有趣的是，我们对问题 (3) 的解决方案为一个多世纪前 Poincaré 提出的问题提供了准线性算法。最后，我们注意到我们针对问题 (1) 的算法扩展到计算两条最长为 ℓ 的曲线的几何交点数○(n+ ℓ2） 时间。