The crossing number of a graph is the minimum number of edge crossings that a graph can have when drawn in the plane. Determining this number, known as the Crossing Number problem, is a celebrated problem in combinatorial optimization. It has been known to be NP-complete since the 1980s, and already showing its fixed-parameter tractability when parameterized by the vertex cover number required fairly involved techniques. In this paper, we prove that computing the crossing number exactly remains NP-hard even for graphs of path-width 12 (and as a result, for simple graphs of path-width 13 and tree-width 9). These results highlight that, although both path- and tree-decompositions have been highly successful tools in many graph algorithm scenarios, general crossing number computation is unlikely (under P $\neq$ NP) to be successfully tackled using graph decompositions of bounded width -- a question that had remained a 'tantalizing open problem' [S. Cabello, Hardness of Approximation for Crossing Number, 2013] till now.

翻译：图的交叉数是将其绘制在平面上时所能达到的最小边交叉数量。确定这一数值的问题（即交叉数问题）是组合优化中一个著名问题。自20世纪80年代起，该问题已被确认为NP完全问题，而即使在以顶点覆盖数为参数证明其固定参数可解性时，也需要相当复杂的技术。本文证明，即使对于路径宽度为12的图（进而，对于路径宽度为13、树宽度为9的简单图），精确计算交叉数仍然是NP难的。这些结果凸显出：尽管路径分解和树分解在许多图算法场景中已成为非常成功的工具，但一般而言，利用有界宽度的图分解来求解交叉数计算问题（在P≠NP的假设下）很可能无法成功——这一问题此前一直是一个“诱人的开放问题”[S. Cabello, Hardness of Approximation for Crossing Number, 2013]，直到现在。