Galois Connections for Patterns: An Algebra of Labelled Graphs

David A Cohen,Stanislav Živný,Peter G Jeavons,Martin C Cooper

doi:10.1007/978-3-030-72308-8_9

Abstract

AbstractA pattern is a generic instance of a binary constraint satisfaction problem (CSP) in which the compatibility of certain pairs of variable-value assignments may be unspecified. The notion of forbidden pattern has led to the discovery of several novel tractable classes for the CSP. However, for this field to come of age it is time for a theoretical study of the algebra of patterns. We present a Galois connection between lattices composed of sets of forbidden patterns and sets of generic instances, and investigate its consequences. We then extend patterns to augmented patterns and exhibit a similar Galois connection. Augmented patterns are a more powerful language than flat (i.e. non-augmented) patterns, as we demonstrate by showing that, for any $$k \ge 1$$ k ≥ 1 , instances with tree-width bounded by k cannot be specified by forbidding a finite set of flat patterns but can be specified by a finite set of augmented patterns. A single finite set of augmented patterns can also describe the class of instances such that each instance has a weak near-unanimity polymorphism of arity k (thus covering all tractable language classes).We investigate the power of forbidding augmented patterns and discuss their potential for describing new tractable classes.

Highlights

The constraint satisfaction problem (CSP) (Constraint Satisfaction Problem) is a classical abstract framework for the modelling of finite-domain constrained assignment problems [8,32]
In this paper we have initiated the study of the Galois connection between lattices of sets of forbidden patterns and sets of instances
We have seen that both bounded treewidth and the existence of a polymorphism satisfying a set of identities can be expressed using augmented patterns

Summary

Introduction

The CSP (Constraint Satisfaction Problem) is a classical abstract framework for the modelling of finite-domain constrained assignment problems [8,32]. We begin by studying what we call flat patterns before studying augmented patterns with extra structure, such as partial orders on variables or domain values Adding such structure is essential to define certain hybrid classes such as BTP [16] and EMC [19], but, as we will show, allows us to define (families of) polymorphisms [28] and bounded tree-width [20] within the same framework. For both flat and augmented patterns, we exhibit a Galois connection between sets of patterns and sets of instances.

Definitions and Notation

The Two Lattices

The Galois Connection

Tractability Consequences of the Galois Connection

Motivation

Augmented Patterns

A Galois Connection for Augmented Patterns

Discussion and Conclusion