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

Authors:
Christoph Buchheim;Markus Chimani;Dietmar Ebner;Carsten Gutwenger;Michael JüNger;Gunnar W. Klau;Petra Mutzel;René Weiskircher
Affiliations:
Department of Computer Science, University of Cologne, Germany;Department of Computer Science, University of Dortmund, Germany;Institute of Computer Languages, Vienna University of Technology, Austria;Department of Computer Science, University of Dortmund, Germany;Department of Computer Science, University of Cologne, Germany;Department of Mathematics and Computer Science, Free University Berlin, Germany and DFG Research Center Matheon, Germany;Department of Computer Science, University of Dortmund, Germany;CSIRO Mathematical and Information Sciences, Melbourne, Australia
Venue:
Discrete Optimization
Year:
2008

Citing 23
Cited 7

Computer aided layout of entity relationship diagrams

Journal of Systems and Software - Special double issue on the entity-relationship approach to databases and related software
Crossing Minimization in Linear Embeddings of Graphs

IEEE Transactions on Computers
String graphs. II.: Recognizing string graphs is NP-hard

Journal of Combinatorial Theory Series B
An O(m log n)-time algorithm for the maximal planar subgraph problem

SIAM Journal on Computing
Alpha-algorithms for incremental planarity testing (preliminary version)

STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
On-Line Planarity Testing

SIAM Journal on Computing
An experimental comparison of four graph drawing algorithms

Computational Geometry: Theory and Applications
A better approximation algorithm for finding planar subgraphs

Journal of Algorithms
On integrating constraint propagation and linear programming for combinatorial optimization

AAAI '99/IAAI '99 Proceedings of the sixteenth national conference on Artificial intelligence and the eleventh Innovative applications of artificial intelligence conference innovative applications of artificial intelligence
Efficient Planarity Testing

Journal of the ACM (JACM)
The ABACUS system for branch-and-cut-and-price algorithms in integer programming and combinatorial optimization

Software—Practice & Experience - Special issue on discrete algorithm engineering
Computing crossing numbers in quadratic time

STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Computers and Intractability: A Guide to the Theory of NP-Completeness

Computers and Intractability: A Guide to the Theory of NP-Completeness
A Linear Algorithm for the Maximal Planar Subgraph Problem

WADS '95 Proceedings of the 4th International Workshop on Algorithms and Data Structures
Which Aesthetic has the Greatest Effect on Human Understanding?

GD '97 Proceedings of the 5th International Symposium on Graph Drawing
A Linear Time Implementation of SPQR-Trees

GD '00 Proceedings of the 8th International Symposium on Graph Drawing
Improved Bounds for the Crossing Numbers of Km,n and Kn

SIAM Journal on Discrete Mathematics
Reduction of symmetric semidefinite programs using the regular $$\ast$$-representation

Mathematical Programming: Series A and B
Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms

Journal of Computer and System Sciences
Non-planar core reduction of graphs

GD'05 Proceedings of the 13th international conference on Graph Drawing
A linear time algorithm for finding a maximal planar subgraph based on PC-trees

COCOON'05 Proceedings of the 11th annual international conference on Computing and Combinatorics
A note on computing a maximal planar subgraph using PQ-trees

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems
O(n2) algorithms for graph planarization

IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems

A New Approach to Exact Crossing Minimization

ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Inserting a vertex into a planar graph

SODA '09 Proceedings of the twentieth Annual ACM-SIAM Symposium on Discrete Algorithms
Experiments on exact crossing minimization using column generation

Journal of Experimental Algorithmics (JEA)
Graph drawing algorithms

Algorithms and theory of computation handbook
Facets in the Crossing Number Polytope

SIAM Journal on Discrete Mathematics
Experiments on exact crossing minimization using column generation

WEA'06 Proceedings of the 5th international conference on Experimental Algorithms
Improving the accuracy of linear programming solvers with iterative refinement

Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

Quantified Score

Hi-index	0.00

Visualization

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"1"1 or K"9","1"1 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 for the first time.