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Journal of Combinatorial Optimization

, Volume 33, Issue 4, pp 1183–1225 | Cite as

A tighter insertion-based approximation of the crossing number

  • Markus Chimani
  • Petr Hliněný
Article

Abstract

Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of \(G+F\) with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. Finding an exact solution to MEI is NP-hard for general F. We present the first polynomial time algorithm for MEI that achieves an additive approximation guarantee—depending only on the size of F and the maximum degree of G, in the case of connected G. Our algorithm seems to be the first directly implementable one in that realm, too, next to the single edge insertion. It is also known that an (even approximate) solution to the MEI problem would approximate the crossing number of the F-almost-planar graph \(G+F\), while computing the crossing number of \(G+F\) exactly is NP-hard already when \(|F|=1\). Hence our algorithm induces new, improved approximation bounds for the crossing number problem of F-almost-planar graphs, achieving constant-factor approximation for the large class of such graphs of bounded degrees and bounded size of F.

Keywords

Planar graph Multiple edge insertion SPQR tree Crossing number 

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Theoretical Computer ScienceUniversity OsnabrückOsnabrückGermany
  2. 2.Faculty of InformaticsMasaryk UniversityBrnoCzech Republic

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