Doubly lexical orderings of matrices
SIAM Journal on Computing
Breaking Row and Column Symmetries in Matrix Models
CP '02 Proceedings of the 8th International Conference on Principles and Practice of Constraint Programming
Constraint Models for the Covering Test Problem
Constraints
Symmetry Breaking using Value Precedence
Proceedings of the 2006 conference on ECAI 2006: 17th European Conference on Artificial Intelligence August 29 -- September 1, 2006, Riva del Garda, Italy
The complexity of global constraints
AAAI'04 Proceedings of the 19th national conference on Artifical intelligence
IJCAI'03 Proceedings of the 18th international joint conference on Artificial intelligence
Breaking symmetries in all different problems
IJCAI'05 Proceedings of the 19th international joint conference on Artificial intelligence
CP'07 Proceedings of the 13th international conference on Principles and practice of constraint programming
Modelling equidistant frequency permutation arrays: an application of constraints to mathematics
CP'09 Proceedings of the 15th international conference on Principles and practice of constraint programming
Snake lex: an alternative to double lex
CP'09 Proceedings of the 15th international conference on Principles and practice of constraint programming
Symmetry within and between solutions
PRICAI'10 Proceedings of the 11th Pacific Rim international conference on Trends in artificial intelligence
Branch-cut-and-propagate for the maximum k-colorable subgraph problem with symmetry
CPAIOR'11 Proceedings of the 8th international conference on Integration of AI and OR techniques in constraint programming for combinatorial optimization problems
Automatic generation of constraints for partial symmetry breaking
CP'11 Proceedings of the 17th international conference on Principles and practice of constraint programming
Symmetry breaking via LexLeader feasibility checkers
IJCAI'11 Proceedings of the Twenty-Second international joint conference on Artificial Intelligence - Volume Volume One
Increasing symmetry breaking by preserving target symmetries
CP'12 Proceedings of the 18th international conference on Principles and Practice of Constraint Programming
The regulargcc matrix constraint
AI'12 Proceedings of the 25th Australasian joint conference on Advances in Artificial Intelligence
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We consider a common type of symmetry where we have a matrix of decision variables with interchangeable rows and columns. A simple and efficient method to deal with such row and column symmetry is to post symmetry breaking constraints like DOUBLELEX and SNAKELEX. We provide a number of positive and negative results on posting such symmetry breaking constraints. On the positive side, we prove that we can compute in polynomial time a unique representative of an equivalence class in a matrix model with row and column symmetry if the number of rows (or of columns) is bounded and in a number of other special cases. On the negative side, we show that whilst DOUBLELEX and SNAKELEX are often effective in practice, they can leave a large number of symmetric solutions in the worst case. In addition, we prove that propagating DOUBLELEX completely is NP-hard. Finally we consider how to break row, column and value symmetry, correcting a result in the literature about the safeness of combining different symmetry breaking constraints. We end with the first experimental study on how much symmetry is left by DOUBLELEX and SNAKELEX on some benchmark problems.