An Optimal Arc Consistency Algorithm for a Chain of Atmost Constraints with Cardinality

Abstract

Abstract. The ATMOSTSEQCARD constraint is the conjunction of a cardinality constraint on a sequence of n variables and of n − q + 1 constraints ATMOST u on each subsequence of size q. This constraint is useful in car-sequencing and crew-rostering problems. In [18], two algorithms designed for the AMONGSEQ constraint were adapted to this constraint with a O(2 q n) and O(n 3 ) worst case time complexity, respectively. In [10], another algorithm with a O(n 2 log n) worst case time complexity and similarly adaptable to filter ATMOSTSEQCARD in O(n log n) was proposed. In this paper, we introduce an algorithm for achieving Arc Consistency on the ATMOSTSEQCARD constraint with a O(n) (hence optimal) worst case time complexity. We then empirically study the efficiency of our propagator on instances of the car-sequencing and crew-rostering problems.

SAT and Hybrid Models of the Car Sequencing Problem

“…In [21], the authors proposed a O(n) filtering algorithm achieving AC on this constraint. We outline the main idea of the propagator.…”

“…Now we have all the tools to define the propagator associated to the constraint ATMOSTSEQCARD described in [21], and which is a conjunction of GAC on the ATMOST constraints on each subsequence, of CARDINALITY constraint…”

“…Overall efficiency We first observe that most of the approaches we introduce in this paper significantly improve the state of the art, at least for systematic methods. For instance, in the experiments reported in [21] several instances of the set sat [hard] were not solved within a 20 minutes cutoff. Moreover we are not aware of other systematic approaches being able to solve these instances.…”

“…capacity constraints). Most recently, The ATMOSTSEQCARD constraint [21] or its combination with the Global Sequencing Constraint (GSC) [19] showed outstanding results compared to other CP models. However, based on experiments, pure CP approaches suffer when the task becomes proving unsatisfiability.…”

SAT Encodings for the Car Sequencing Problem

“…Over the years different approaches have been used to solve these instances, among them constraint programming, local search and integer programming [14][10] [11][6] [15].…”