Structural Tractability of Propagated Constraints

Abstract

Modern constraint solvers do not require constraints to be represented using any particular data structure. Instead, constraints are given as black boxes known as propagators. Propagators are given a list of current domains for variables and are allowed to prune values not consistent with these current domains.

Using propagation as the only primitive operation on constraints imposes restrictions on the operations that can be performed in polynomial time. In the extensional representation of constraints (so-called positive table constraints) join and project are primitive polynomial-time operations. This is not true for propagated constraints.

The question we pose in this paper is: If propagation is the only primitive operation, what are the structurally tractable classes of constraint programs (whose instances can be solved in polynomial time)?

We consider a hierarchy of propagators: arbitrary propagators, whose only ability is consistency checking; partial assignment membership propagators, which allow us to check partial assignments; and generalised arc consistency propagators, the strongest form of propagator.

In the first two cases, we answer the posed question by establishing dichotomies. In the case of generalised arc consistency propagators, we achieve a useful dichotomy in the restricted case of acyclic structures.