A branch-and-cut approach to the crossing number problem - Kölner UniversitätsPublikationsServer

Buchheim, Christoph ORCID: 0000-0001-9974-404X, Chimani, Markus, Ebner, Dietmar, Gutwenger, Carsten, Klau, Gunnar W., Mutzel, Petra ORCID: 0000-0001-7621-971X and Weiskircher, René (2008). A branch-and-cut approach to the crossing number problem. Discrete optimization, 5 (2). pp. 373-388. Elsevier. Open Access

Preview

PDF
zaik2006-508.pdf - Submitted Version
Download (430kB) | Preview

Abstract

The crossing number of a graph is the minimum number of edge crossings in any drawing of the graph in the plane. Extensive research has produced bounds on the crossing number and exact formulae for special graph classes, yet the crossing numbers of graphs such as K_{11} or K_{9,11} are still unknown. Finding the crossing number is NP-hard for general graphs and no practical algorithm for its computation has been published so far. We present an integer linear programming formulation that is based on a reduction of the general problem to a restricted version of the crossing number problem in which each edge may be crossed at most once. We also present cutting plane generation heuristics and a column generation scheme. As we demonstrate in a computational study, a branch-and-cut algorithm based on these techniques as well as recently published preprocessing algorithms can be used to successfully compute the crossing number for small to medium sized general graphs.

Item Type:

Journal Article

Creators:

Creators	Email	ORCID	ORCID Put Code
Buchheim, Christoph	UNSPECIFIED	orcid.org/0000-0001-9974-404X	UNSPECIFIED
Chimani, Markus	UNSPECIFIED	UNSPECIFIED	UNSPECIFIED
Ebner, Dietmar	UNSPECIFIED	UNSPECIFIED	UNSPECIFIED
Gutwenger, Carsten	UNSPECIFIED	UNSPECIFIED	UNSPECIFIED
Klau, Gunnar W.	UNSPECIFIED	UNSPECIFIED	UNSPECIFIED
Mutzel, Petra	UNSPECIFIED	orcid.org/0000-0001-7621-971X	UNSPECIFIED
Weiskircher, René	UNSPECIFIED	UNSPECIFIED	UNSPECIFIED