The power of propagation: when GAC is enough

Abstract

Considerable effort in constraint programming has focused on the development of efficient propagators for individual constraints. In this paper, we consider the combined power of such propagators when applied to collections of more than one constraint. In particular we identify classes of constraint problems where such propagators can decide the existence of a solution on their own, without the need for any additional search. Sporadic examples of such classes have previously been identified, including classes based on restricting the structure of the problem, restricting the constraint types, and some hybrid examples. However, there has previously been no unifying approach which characterises all of these classes: structural, language-based and hybrid. In this paper we develop such a unifying approach and embed all the known classes into a common framework. We then use this framework to identify a further class of problems that can be solved by propagation alone.

Using Constraint Propagation to Bound Linear Programs

“…It is of interest in the constraint programming community to identify subsets of problems where enforcing a precise form of local consistency is a refutation-complete method, i.e., where attaining the local consistency implies satisfiability of the original system. This was studied, e.g., for arc consistency by Cohen and Jeavons (2017) and Cooper and Živný (2016). Dlask and Werner (2022) identified two classes of problems where activity propagation is refutation complete (more precisely, where BCD applied to the dual linear program is optimal, which is equivalent to refutation-completeness of activity propagation by Remark 7).…”

Constraint Reasoning

“…As a result, MAC finds a solution backtrack-free, that is, in polynomial time if all constraints can be made arc consistent in polynomial time. Cohen and Jeavons [2017] recently showed that the condition is tight because Berge-acyclicity is the only structural property that allows arc consistency to decide satisfiability. Berge-acyclicity is thus the only structural property that allows MAC to find a solution in polynomial time.…”

On Singleton Arc Consistency for CSPs Defined by Monotone Patterns

“…Of key importance amongst these pre-processing algorithms are the relatives of arc consistency propagation including generalised arc consistency (GAC) and singleton arc consistency (SAC). Surprisingly there are large classes [16,23,13,28] of the CSP for which GAC or SAC are decision procedures: after establishing consistency if every variable still has a non-empty domain then the instance has a solution.…”

“…Of key importance amongst these pre-processing algorithms are the relatives of arc consistency propagation including generalised arc consistency (GAC) and singleton arc consistency (SAC). Surprisingly there are large classes [16,23,13,28] of the CSP for which GAC or SAC are decision procedures: after establishing consistency if every variable still has a non-empty domain then the instance has a solution.More generally, these results fit into the wider area of research aiming to identify subproblems of the CSP for which certain polynomial-time algorithms are decision procedures. Perhaps the most natural ways to restrict the CSP is to limit the constraint relations that we allow or to limit the structure of (the hypergraph of) interactions of the constraint scopes.…”