Structural Decompositions for Problems with Global Constraints

TL;DR

This paper investigates how structural restrictions can make solving CSPs with global constraints more tractable, especially when the number of solutions is limited, extending to cost-based constraints.

Contribution

It demonstrates that bounded solution counts enable tractability of CSPs with global constraints under structural restrictions, expanding known tractable classes.

Findings

Structural restrictions can yield tractable classes with global constraints.

Bounded solution counts are key to tractability.

Results extend to cost-assigning constraints.

Abstract

A wide range of problems can be modelled as constraint satisfaction problems (CSPs), that is, a set of constraints that must be satisfied simultaneously. Constraints can either be represented extensionally, by explicitly listing allowed combinations of values, or implicitly, by special-purpose algorithms provided by a solver. Such implicitly represented constraints, known as global constraints, are widely used; indeed, they are one of the key reasons for the success of constraint programming in solving real-world problems. In recent years, a variety of restrictions on the structure of CSP instances have been shown to yield tractable classes of CSPs. However, most such restrictions fail to guarantee tractability for CSPs with global constraints. We therefore study the applicability of structural restrictions to instances with such constraints. We show that when the number of solutions to a CSP instance is bounded in key parts of the problem, structural restrictions can be used to derive new tractable classes. Furthermore, we show that this result extends to combinations of instances drawn from known tractable classes, as well as to CSP instances where constraints assign costs to satisfying assignments.