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.
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.
@cite_18
1987681136
We develop new techniques for deriving very strong computational lower bounds for a class of well-known NP-hard problems, including weighted satisfiability , dominating set , hitting set , set cover , clique , and independent set . For example, although a trivial enumeration can easily test in time O(nk) if a given graph of n vertices has a clique of size k, we prove that unless an unlikely collapse occurs in parameterized complexity theory, the problem is not solvable in time f(k) no(k) for any function f, even if we restrict the parameter value k to be bounded by an arbitrarily small function of n. Under the same assumption, we prove that even if we restrict the parameter values k to be Θ(μ(n)) for any reasonable function μ, no algorithm of running time no(k) can test if a graph of n vertices has a clique of size k. Similar strong lower bounds are also derived for other problems in the above class. Our techniques can be extended to derive computational lower bounds on approximation algorithms for NP-hard optimization problems. For example, we prove that the NP-hard distinguishing substring selection problem, for which a polynomial time approximation scheme has been recently developed, has no polynomial time approximation schemes of running time f(1 e)no(1 e) for any function f unless an unlikely collapse occurs in parameterized complexity theory.