Crossing Number is Hard for Kernelization

TL;DR

This paper proves that the crossing number problem does not admit a polynomial kernel, using cross-composition techniques, and establishes NP-hardness for the tile crossing number problem of twisted planar tiles.

Contribution

It demonstrates the non-existence of polynomial kernels for crossing number, and proves NP-hardness of the tile crossing number problem, resolving open questions in graph drawing complexity.

Findings

Crossing number problem has no polynomial kernel unless NP ⊆ coNP/poly.

Tile crossing number problem of twisted planar tiles is NP-hard.

Results hold even for graphs derived from planar graphs by adding one edge.

Abstract

The graph crossing number problem, cr(G)<=k, asks for a drawing of a graph G in the plane with at most k edge crossings. Although this problem is in general notoriously difficult, it is fixed- parameter tractable for the parameter k [Grohe]. This suggests a closely related question of whether this problem has a polynomial kernel, meaning whether every instance of cr(G)<=k can be in polynomial time reduced to an equivalent instance of size polynomial in k (and independent of |G|). We answer this question in the negative. Along the proof we show that the tile crossing number problem of twisted planar tiles is NP-hard, which has been an open problem for some time, too, and then employ the complexity technique of cross-composition. Our result holds already for the special case of graphs obtained from planar graphs by adding one edge.