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

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.

[1]  János Pach,et al.  Graphs drawn with few crossings per edge , 1996, GD.

[2]  John N. Hooker,et al.  On Integrating Constraint Propagation and Linear Programming for Combinatorial Optimization , 1999, AAAI/IAAI.

[3]  Alexander Schrijver,et al.  Reduction of symmetric semidefinite programs using the regular $$\ast$$-representation , 2007, Math. Program..

[4]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[5]  W. Sierpinski,et al.  Sur le probléme des courbes gauches en Topologie , 2022 .

[6]  Jan Kratochvíl,et al.  String graphs. II. recognizing string graphs is NP-hard , 1991, J. Comb. Theory, Ser. B.

[7]  Roberto Tamassia,et al.  On-line maintenance of triconnected components with SPQR-trees , 1996, Algorithmica.

[8]  Robert E. Tarjan,et al.  An O(m log n)-Time Algorithm for the Maximal Planar Subgraph Problem , 1992, SIAM J. Comput..

[9]  Hristo Djidjev.  A Linear Algorithm for the Maximal Planar Subgraph Problem , 1995, WADS.

[10]  Matteo Fischetti,et al.  Combinatorial Benders' Cuts , 2004, IPCO.

[11]  Paul Turán,et al.  A note of welcome , 1977, J. Graph Theory.

[12]  M.N.S. Swamy,et al.  O(n2) algorithms for graph planarization , 1989, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[13]  Petra Mutzel,et al.  The Constrained Crossing Minimization Problem , 1999, Graph Drawing.

[14]  Cristina G. Fernandes,et al.  A better approximation algorithm for finding planar subgraphs , 1996, SODA '96.

[15]  Michael Jünger,et al.  Maximum planar subgraphs and nice embeddings: Practical layout tools , 1996, Algorithmica.

[16]  Serge A. Plotkin,et al.  Routing and admission control in general topology networks with Poisson arrivals , 1995, SODA '96.

[17]  Kazimierz Zarankiewicz,et al.  The Solution of a Certain Problem on graphs of P. Turan , 1953 .

[18]  Peter Eades,et al.  Edge crossings in drawings of bipartite graphs , 1994, Algorithmica.

[19]  Giuseppe Liotta,et al.  An Experimental Comparison of Four Graph Drawing Algorithms , 1997, Comput. Geom..

[20]  Petra Mutzel,et al.  Inserting an Edge into a Planar Graph , 2001, SODA '01.

[21]  Robert E. Tarjan,et al.  Computing an st -Numbering , 1976, Theor. Comput. Sci..

[22]  Etienne de Klerk,et al.  Improved Bounds for the Crossing Numbers of Km, n and Kn , 2004, SIAM J. Discret. Math..

[23]  Petra Mutzel,et al.  An Experimental Study of Crossing Minimization Heuristics , 2003, Graph Drawing.

[24]  Michael Jünger,et al.  A note on computing a maximal planar subgraph using PQ-trees , 1998, IEEE Trans. Comput. Aided Des. Integr. Circuits Syst..

[25]  La Poutre,et al.  Alpha-algorithms for incremental planarity testing , 1994 .

[26]  Michael Jünger,et al.  The ABACUS system for branch‐and‐cut‐and‐price algorithms in integer programming and combinatorial optimization , 2000, Softw. Pract. Exp..

[27]  Martin Grohe.  Computing crossing numbers in quadratic time , 2004, J. Comput. Syst. Sci..

[28]  Robert E. Tarjan,et al.  Dividing a Graph into Triconnected Components , 1973, SIAM J. Comput..

[29]  Frank Thomson Leighton,et al.  A Framework for Solving VLSI Graph Layout Problems , 1983, J. Comput. Syst. Sci..

[30]  Roberto Tamassia,et al.  On-Line Planarity Testing , 1989, SIAM J. Comput..

[31]  Han La Poutré,et al.  Alpha-algorithms for incremental planarity testing (preliminary version) , 1994, STOC '94.

[32]  Helen C. Purchase,et al.  Which Aesthetic has the Greatest Effect on Human Understanding? , 1997, GD.

[33]  R. Guy.  Crossing numbers of graphs , 1972 .

[34]  David S. Johnson,et al.  Crossing Number is NP-Complete , 1983 .

[35]  TamassiaRoberto,et al.  On-Line Planarity Testing , 1996 .

[36]  Wen-Lian Hsu,et al.  A Linear Time Algorithm for Finding a Maximal Planar Subgraph Based on PC-Trees , 2005, COCOON.

[37]  Patrice Ossona de Mendez,et al.  On Cotree-Critical and DFS Cotree-Critical Graphs , 2003, J. Graph Algorithms Appl..

[38]  Markus Chimani,et al.  Non-planar core reduction of graphs , 2009, Discret. Math..

[39]  Kellogg S. Booth,et al.  Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-Tree Algorithms , 1976, J. Comput. Syst. Sci..

[40]  Petra Mutzel,et al.  A Linear Time Implementation of SPQR-Trees , 2000, GD.

[41]  Sumio Masuda,et al.  Crossing Minimization in Linear Embeddings of Graphs , 1990, IEEE Trans. Computers.

[42]  George B. Dantzig,et al.  Decomposition Principle for Linear Programs , 1960 .

[43]  Alexander Grigoriev,et al.  Algorithms for Graphs Embeddable with Few Crossings per Edge , 2005, Algorithmica.

[44]  Carlo Batini,et al.  Computer aided layout of entity relationship diagrams , 1984, J. Syst. Softw..

[45]  Robert E. Tarjan,et al.  Efficient Planarity Testing , 1974, JACM.