Symmetry Breaking Revisited

Symmetries in constraint satisfaction problems (CSPs) are one of the difficulties that practitioners have to deal with. We present in this paper a new method based on the symmetries of decisions taken from the root of the search tree. This method can be seen as an improvement of the nogood recording presented by Focacci and Milano[5] and Fahle, Schamberger and Sellmann[4]. We present a simple formalization of our method for which we prove correctness and completeness results. We also show that our method is theoretically more efficient as the number of dominance checks, the number of nogoods and the size of each nogood are smaller. This is confirmed by an experimental evaluation on the social golfer problem, a very difficult and highly symmetrical real world problem. We are able to break all symmetries for problems with more than 1036 symmetries. We report both new results, and a comparison with previous work.

[1]  Meinolf Sellmann,et al.  Heuristic Constraint Propagation , 2002, CP.

[2]  Carmen Gervet,et al.  Interval propagation to reason about sets: Definition and implementation of a practical language , 1997, Constraints.

[3]  Barbara M. Smith,et al.  Partial Symmetry Breaking , 2002, CP.

[4]  Justin Pearson.  Symmetry Breaking in Constraint Satisfaction with Graph-Isomorphism: Comma-Free Codes , 2004, ISAIM.

[5]  François Pachet,et al.  Using Symmetry of Global Constraints to Speed up the Resolution of Constraint Satisfaction Problems , 2002 .

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

[7]  Rolf Backofen,et al.  Excluding Symmetries in Constraint-Based Search , 1999, CP.

[8]  Pascal Brisset,et al.  Solving Kirkman’s Schoolgirl Problem in a Few Seconds , 2004, Constraints.

[9]  Jean-François Puget,et al.  Symmetry Breaking Revisited , 2002, Constraints.

[10]  Pascal Van Hentenryck,et al.  Tractable Symmetry Breaking for CSPs with Interchangeable Values , 2003, IJCAI.

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

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

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

[14]  Ian P. Gent,et al.  Symmetry Breaking in Constraint Programming , 2000, ECAI.

[15]  Carme Torras,et al.  Exploiting symmetries within constraint satisfaction search , 2001, Artif. Intell..

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

[17]  Michela Milano,et al.  Global Cut Framework for Removing Symmetries , 2001, CP.

[18]  Meinolf Sellmann,et al.  Symmetry Breaking , 2001, CP.

[19]  Jean-François Puget,et al.  On the Satisfiability of Symmetrical Constrained Satisfaction Problems , 1993, ISMIS.