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Crossing Number is Hard for Kernelization

Authors Petr Hlinený, Marek Dernár

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Petr Hlinený
Marek Dernár

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Petr Hlinený and Marek Dernár. Crossing Number is Hard for Kernelization. In 32nd International Symposium on Computational Geometry (SoCG 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 51, pp. 42:1-42:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.SoCG.2016.42

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, STOC 2001]. 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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Keywords
  • crossing number; tile crossing number; parameterized complexity; polynomial kernel; cross-composition

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