Detecting Symmetries by Branch & Cut

We present a new approach for detecting automorphisms and symmetries of an arbitrary graph based on branch & cut. We derive an IP-model for this problem and have a first look on cutting planes and primal heuristics. The algorithm was implemented within the ABACUS framework; its experimental runtimes are promising.

[1]  Sang Ho Lee,et al.  Finding Planar Geometric Automorphisms in Planar Graphs , 1998, ISAAC.

[2]  Hubert de Fraysseix,et al.  An Heuristic for Graph Symmetry Detection , 1999, GD.

[3]  L. Lovász,et al.  Geometric Algorithms and Combinatorial Optimization , 1981 .

[4]  M. R. Rao,et al.  Odd Minimum Cut-Sets and b-Matchings , 1982, Math. Oper. Res..

[5]  Michel X. Goemans,et al.  Minimizing submodular functions over families of sets , 1995, Comb..

[6]  Anna Lubiw,et al.  Some NP-Complete Problems Similar to Graph Isomorphism , 1981, SIAM J. Comput..

[7]  M. Balinski,et al.  On the Assignment Polytope , 1974 .

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

[9]  Brendan D. McKay,et al.  Symmetric drawings of triconnected planar graphs , 2002, SODA '02.

[10]  Derek G. Corneil,et al.  The graph isomorphism disease , 1977, J. Graph Theory.

[11]  Eugene L. Lawler,et al.  The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization , 1985 .

[12]  Egon Balas.  The Asymmetric Assignment Problem and Some New Facets of the Traveling Salesman Polytope on a Directed Graph , 1989, SIAM J. Discret. Math..

[13]  E. Wright,et al.  An Introduction to the Theory of Numbers , 1939 .

[14]  Xuemin Lin,et al.  Spring algorithms and symmetry , 2000, Theor. Comput. Sci..

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

[16]  András A. Benczúr,et al.  Fast Algorithms for Even/Odd Minimum Cuts and Generalizations , 2000, ESA.

[17]  Joseph Manning,et al.  Computational Complexity of Geometric Symmetry Detection in Graphs , 1989, Great Lakes Computer Science Conference.

[18]  Donald E. Taylor,et al.  A Group-Theoretic Method for Drawing Graphs Symmetrically , 2002, Graph Drawing.

[19]  Egon Balas,et al.  A lifting procedure for the asymmetric traveling salesman polytope and a large new class of facets , 1993, Math. Program..