Automatic Detection of Variable and Value Symmetries

Many symmetry breaking techniques assume that the symmetries of a CSP are given as input in addition to the CSP itself. We present a method that can be used to detect all the symmetries of a CSP. This method constructs a graph that has the same symmetries as the CSP. Then, generators for the symmetry group are computed using a graph automorphism algorithm. This method improves and extends previous work in order to cover global constraints, arithmetic expressions and value symmetries. We show that this method is able to find symmetries for examples that were thought to be too convoluted for automated detection. We also show that the overhead of symmetry detection is quite negligible, even on very large instances. We present a comprehensive set of examples where automated symmetry detection is coupled with symmetry breaking techniques.

[1]  Jacobo Torán,et al.  The graph isomorphism problem , 2020, Commun. ACM.

[2]  Pascal Van Hentenryck,et al.  Structural Symmetry Breaking , 2005, IJCAI.

[3]  Igor L. Markov,et al.  Automatically Exploiting Symmetries in Constraint Programming , 2004, CSCLP.

[4]  Nicolas Barnier,et al.  Solving the Kirkman's schoolgirl problem in a few seconds , 2002 .

[5]  Jean-François Puget.  Symmetry Breaking Using Stabilizers , 2003, CP.

[6]  Igor L. Markov,et al.  Exploiting structure in symmetry detection for CNF , 2004, Proceedings. 41st Design Automation Conference, 2004..

[7]  Toby Walsh,et al.  Breaking Row and Column Symmetries in Matrix Models , 2002, CP.

[8]  Steve Linton,et al.  Generic SBDD Using Computational Group Theory , 2003, CP.

[9]  Jean-François Puget.  Breaking symmetries in all different problems , 2005, IJCAI.

[10]  Steve Linton,et al.  New Developments in Symmetry Breaking in Search Using Computational Group Theory , 2004, AISC.

[11]  James M. Crawford,et al.  Symmetry-Breaking Predicates for Search Problems , 1996, KR.

[12]  Jean-Charles Régin,et al.  Generalized Arc Consistency for Global Cardinality Constraint , 1996, AAAI/IAAI, Vol. 1.

[13]  François Margot,et al.  Exploiting orbits in symmetric ILP , 2003, Math. Program..

[14]  Toby Walsh,et al.  CGRASS: A System for Transforming Constraint Satisfaction Problems , 2002, International Workshop on Constraint Solving and Constraint Logic Programming.

[15]  Jean-Daniel Zucker,et al.  Abstraction, Reformulation and Approximation, 6th International Symposium, SARA 2005, Airth Castle, Scotland, UK, July 26-29, 2005, Proceedings , 2005, SARA.

[16]  Steve Linton,et al.  Tractable Symmetry Breaking Using Restricted Search Trees , 2004, ECAI.

[17]  Jean-François Puget,et al.  Breaking All Value Symmetries in Surjection Problems , 2005, CP.

[18]  Francesca Rossi,et al.  Principles and Practice of Constraint Programming – CP 2003 , 2003, Lecture Notes in Computer Science.

[19]  Pascal Van Hentenryck,et al.  Compositional Derivation of Symmetries for Constraint Satisfaction , 2005, SARA.

[20]  Warwick Harvey,et al.  Groups and Constraints: Symmetry Breaking during Search , 2002, CP.

[21]  J. Köbler,et al.  The Graph Isomorphism Problem: Its Structural Complexity , 1993 .

[22]  Barbara M. Smith,et al.  Value Ordering for Finding All Solutions , 2005, IJCAI.