Crossing Number is NP-hard for Constant Path-width (and Tree-width)
This result shows that bounded-width graph decompositions are unlikely to yield efficient algorithms for crossing number computation, impacting researchers in graph theory and algorithm design.
The paper proves that computing the crossing number exactly is NP-hard for graphs with constant path-width 12 (and tree-width 9), addressing a long-standing open problem in combinatorial optimization.
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.