A tighter insertion-based approximation of the crossing number

J 2017

A tighter insertion-based approximation of the crossing number

CHIMANI, Markus and Petr HLINĚNÝ

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

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10201 Computer sciences, information science, bioinformatics

Country of publisher

Germany

Confidentiality degree

není předmětem státního či obchodního tajemství

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

Abstract

V originále

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

Citovat

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.

@article{1372354,
   author = {Chimani, Markus and Hliněný, Petr},
   article_number = {4},
   doi = {http://dx.doi.org/10.1007/s10878-016-0030-z},
   keywords = {Planar graph; Multiple edge insertion; SPQR tree; Crossing number},
   language = {eng},
   issn = {1382-6905},
   journal = {Journal of Combinatorial Optimization},
   title = {A tighter insertion-based approximation of the crossing number},
   volume = {33},
   year = {2017}
}

TY  - JOUR
ID  - 1372354
AU  - Chimani, Markus - Hliněný, Petr
PY  - 2017
TI  - A tighter insertion-based approximation of the crossing number
JF  - Journal of Combinatorial Optimization
VL  - 33
IS  - 4
SP  - 1183-1225
EP  - 1183-1225
PB  - Springer
SN  - 13826905
KW  - Planar graph
KW  - Multiple edge insertion
KW  - SPQR tree
KW  - Crossing number
N2  - 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.
ER  -

CHIMANI, Markus and Petr HLINĚNÝ. A tighter insertion-based approximation of the crossing number. \textit{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.

Detailed Information on Publication Record