Foundations of Symmetry Breaking Revisited

Puget’s article “On the Satisfiability of Symmetrical Constraint Satisfaction Problems” [Puget, 1993] is widely acknowledged to be one of the founding papers in symmetry breaking (see for example [Flener et al., 2006; Rossi et al., 2006; Walsh, 2006]). To date, Puget’s definition of a valid reduction seems to be the only common ground that all authors on static symmetry breaking constraints agree on. His original definition of a valid reduction is for a restricted form of variable symmetries. We extend Puget’s definition to the recent, more general definition of symmetries by Cohen et al. [Cohen et al., 2006]. In our extension, we require a valid reduction to be a tighter constraint satisfaction problem instead of one with more constraints. We re-prove both his central theorems on valid reductions for our definition and we prove considerably stronger results on the existence of valid reductions with further properties. Along the way, we present examples which contradict the common believe that we can always eliminate all symmetries with lexicographic ordering constraints.