Binary Constraint Satisfaction Problems Defined by Excluded Topological Minors

TL;DR

This paper introduces a novel approach to defining tractable classes of binary CSPs by forbidding certain patterns as topological minors, extending structural restrictions and capturing language-based restrictions.

Contribution

It extends the concept of pattern restrictions to topological minors in binary CSPs, unifying structural and language restrictions into a single framework.

Findings

Forbidding finite sets of topological minors captures all known tractable classes.

Certain patterns lead to tractable classes only when forbidden as topological minors.

Augmented patterns identify additional tractable classes, including language restrictions.

Abstract

The binary Constraint Satisfaction Problem (CSP) is to decide whether there exists an assignment to a set of variables which satisfies specified constraints between pairs of variables. A binary CSP instance can be presented as a labelled graph encoding both the forms of the constraints and where they are imposed. We consider subproblems defined by restricting the allowed form of this graph. One type of restriction that has previously been considered is to forbid certain specified substructures (patterns). This captures some tractable classes of the CSP, but does not capture classes defined by language restrictions, or the well-known structural property of acyclicity. In this paper we extend the notion of pattern and introduce the notion of a topological minor of a binary CSP instance. By forbidding a finite set of patterns from occurring as topological minors we obtain a compact mechanism for expressing novel tractable subproblems of the binary CSP, including new generalisations of the class of acyclic instances. Forbidding a finite set of patterns as topological minors also captures all other tractable structural restrictions of the binary CSP. Moreover, we show that several patterns give rise to tractable subproblems if forbidden as topological minors but not if forbidden as sub-patterns. Finally, we introduce the idea of augmented patterns that allows for the identification of more tractable classes, including all language restrictions of the binary CSP.