A tighter insertion-based approximation of the crossing number

CHIMANI, Markus and Petr HLINĚNÝ. A tighter insertion-based approximation of the crossing number. Journal of Combinatorial Optimization. Springer, 2017, vol. 33, No 4, p. 1183-1225. ISSN 1382-6905. Available from: https://dx.doi.org/10.1007/s10878-016-0030-z.

Basic information

• Original name: A tighter insertion-based approximation of the crossing number
• Authors: CHIMANI, Markus (40 Austria) and Petr HLINĚNÝ (203 Czech Republic, guarantor, belonging to the institution).
• Edition: Journal of Combinatorial Optimization, Springer, 2017, 1382-6905.

Other information

• Original language: English
• Type of outcome: Article in a journal
• Field of Study: 10201 Computer sciences, information science, bioinformatics
• Country of publisher: Germany
• Confidentiality degree: is not subject to a state or trade secret
• Impact factor: Impact factor: 0.927
• RIV identification code: RIV/00216224:14330/17:00094634
• Organization unit: Faculty of Informatics
• Doi: http://dx.doi.org/10.1007/s10878-016-0030-z
• UT WoS: 000398945100003
• Keywords in English: Planar graph; Multiple edge insertion; SPQR tree; Crossing number
• Tags: formela-journal
• Tags: International impact, Reviewed

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.

Links

• GA14-03501S, research and development project: Name: Parametrizované algoritmy a kernelizace v kontextu diskrétní matematiky a logiky

Investor: Czech Science Foundation