NP-hardness of sortedness constraints
This establishes computational intractability results for constraint programming practitioners working with combinatorial problems involving sorting.
The paper proves that several sortedness constraints in Constraint Programming are NP-hard for integer variables with non-interval domains, including the sort(U,V) constraint and the keysorting(U,V,Keys,P) constraint even under stability conditions.
In Constraint Programming, global constraints allow to model and solve many combinatorial problems. Among these constraints, several sortedness constraints have been defined, for which propagation algorithms are available, but for which the tractability is not settled. We show that the sort(U,V) constraint (Older et. al, 1995) is intractable for integer variables whose domains are not limited to intervals. As a consequence, the similar result holds for the sort(U,V, P) constraint (Zhou, 1996). Moreover, the intractability holds even under the stability condition present in the recently introduced keysorting(U,V,Keys,P) constraint (Carlsson et al., 2014), and requiring that the order of the variables with the same value in the list U be preserved in the list V. Therefore, keysorting(U,V,Keys,P) is intractable as well.