Constraint Satisfaction over Generalized Staircase Constraints

TL;DR

This paper introduces generalized staircase constraints in CSPs, providing new algorithms for arc consistency that improve efficiency, especially for down staircase constraints, and enhances solution methods for these classes.

Contribution

The paper defines generalized staircase constraints, proves arc consistency sufficiency for down staircase CSPs, and develops an optimal algorithm improving solution efficiency.

Findings

Arc consistency suffices for down staircase CSPs.

Proposed O(cd) algorithm for arc consistency is optimal.

Enhanced methods for solving generalized staircase CSPs.

Abstract

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.