Creignou, N., Hermann, M., Krokhin, A., & Salzer, G. (2008). Complexity of Clausal Constraints Over Chains. Theory of Computing Systems, 42(2), 239–255. https://doi.org/10.1007/s00224-007-9003-z
We investigate the complexity of the satisfiability problem of constraints over finite totally ordered domains. In our context, a clausal constraint is a disjunction of inequalities of the form x≥d and x≤d. We classify the complexity of constraints based on clausal patterns. A pattern abstracts away from variables and contains only information about the domain elements and the type of inequalities occurring in a constraint. Every finite set of patterns gives rise to a (clausal) constraint satisfaction problem in which all constraints in instances must have an allowed pattern. We prove that every such problem is either polynomially decidable or NP-complete, and give a polynomial-time algorithm for recognizing the tractable cases. Some of these tractable cases are new and have not been previously identified in the literature.
