text: One of the key research interests in the area of Constraint Satisfaction Problem (CSP) is to identify tractable classes of constraints and develop efficient solutions for them. In this paper, we introduce generalized staircase (GS) constraints which is an important generalization of one such tractable class found in the literature, namely, staircase constraints. GS constraints are of two kinds, down staircase (DS) and up staircase (US). We first examine several properties of GS constraints, and then show that arc consistency is sufficient to determine a solution to a CSP over DS constraints. Further, we propose an optimal O(cd) time and space algorithm to compute arc consistency for GS constraints where c is the number of constraints and d is the size of the largest domain. Next, observing that arc consistency is not necessary for solving a DSCSP, we propose a more efficient algorithm for solving it. With regard to US constraints, arc consistency is not known to be sufficient to determine a solution, and therefore, methods such as path consistency or variable elimination are required. Since arc consistency acts as a subroutine for these existing methods, replacing it by our optimal O(cd) arc consistency algorithm produces a more efficient method for solving a USCSP.
  @cite_18: Abstract Finding solutions to a constraint satisfaction problem is known to be an NP-complete problem in general, but may be tractable in cases where either the set of allowed constraints or the graph structure is restricted. In this paper we identify a restricted set of contraints which gives rise to a class of tractable problems. This class generalizes the notion of a Horn formula in propositional logic to larger domain sizes. We give a polynomial time algorithm for solving such problems, and prove that the class of problems generated by any larger set of constraints is NP-complete.
summary: Referring to the classes of max-closed and min-closed constraints, @cite_18 showed that their solutions can be computed in @math time where @math is the maximum number of supports that a value in a domain can have. In this paper, we show that a constraint is a DS if and only if it is both max-closed and min-closed. Our DS algorithms have significantly lower time complexities for this restricted subclass.