Crossing Number is NP-hard for Constant Path-width (and Tree-width)

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.