Complexity and polymorphisms for digraph constraint problems under some basic constructions

TL;DR

This paper investigates how graph constructions affect the complexity and polymorphism properties of directed graph constraint satisfaction problems, revealing stability and collapse phenomena, and providing a complete characterization for certain digraphs.

Contribution

It analyzes the stability of CSP complexity and polymorphisms under graph constructions, and characterizes polymorphism properties for specific digraph classes.

Findings

CSPs over directed graphs with a total source and sink are solvable by the few subpowers algorithm iff solvable by local consistency.

Strict width and few subpowers solvability are unstable under first order reductions.

Complete characterization of polymorphism properties for digraphs with symmetric closure as a complete graph.

Abstract

The role of polymorphisms in determining the complexity of constraint satisfaction problems is well established. In this context we study the stability of CSP complexity and polymorphism properties under some basic graph theoretic constructions. As applications we observe a collapse in the applicability of algorithms for CSPs over directed graphs with both a total source and a total sink: the corresponding CSP is solvable by the "few subpowers algorithm" if and only if it is solvable by a local consistency check algorithm. Moreover, we find that the property of "strict width" and solvability by few subpowers are unstable under first order reductions. The analysis also yields a complete characterisation of the main polymorphism properties for digraphs whose symmetric closure is a complete graph.