Sequence constraints are useful in a number of applications. Constraints of this class enforce upper and/or lower bounds on all sub-sequences of variables of a given length within a main sequence. For instance, in Crew-Rostering problems, we may want to have an upper bound on the number of worked days in every sub-sequence to meet working regulations. Several constraints of this class have been studied in the Constraint Programming (CP) literature such as GenSequence, AmongSeq, etc. An even more general constraint, Regular, can be used to enforce arbitrary patterns on all sub-sequences. However, the more general a constraint is, the higher is the complexity of reasoning about it. We therefore investigate the AtMostSeqCard constraint, a particular case where the sequence of variables is subject to a chain of AtMost constraints (all subsequences of size q have at most u values in a set v) along with a global cardinality. This constraint is useful in Car-Sequencing and Crew-Rosterning problems. In [van Hoeve et al.], two algorithms designed for the AmongSeq constraint were adapted to this constraint with an O(2^q . n) and O(\n^3) worst case time complexity, respectively. In [Maher et al.], another algorithm similarly adaptable to filter the AtMostSeqCard constraint with a time complexity of O(n^2) was proposed.

We summarize our contributions regarding this constraint in three different axis. We first introduce an algorithm for achieving arc consistency on the AtMostSeqCard constraint with an O(n) (hence optimal) worst case time complexity. Next, we present practical extensions of this constraint and show how the propagator scales accordingly. We finally propose a novel linear time mechanism for explaining this constraint in a hybrid CP/SAT context.
Our experiments show significant improvements on Car-sequencing and Crew-Rostering problems.