Crossing Number is Hard for Kernelization

Petr Hliněný; Marek Derňár

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arXiv · 1512.02379

Crossing Number is Hard for Kernelization

Petr Hliněný·Marek Derňár

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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.

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Petr Hliněný, Marek Derňár. 2016-02-18. Crossing Number is Hard for Kernelization. https://arxiv.org/abs/1512.02379

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