A finer reduction of constraint problems to digraphs

TL;DR

This paper introduces a refined method to reduce general constraint satisfaction problems to digraphs, preserving complexity and polymorphism properties, thus linking key conjectures in CSP theory to digraphs.

Contribution

It presents a new construction that makes the CSP over a structure logspace equivalent to that over a digraph, maintaining polymorphism properties.

Findings

The reduction is logspace equivalent to the original problem.

Polymorphism properties are preserved in the reduction.

Key CSP conjectures are equivalent when restricted to digraphs.

Abstract

It is well known that the constraint satisfaction problem over a general relational structure A is polynomial time equivalent to the constraint problem over some associated digraph. We present a variant of this construction and show that the corresponding constraint satisfaction problem is logspace equivalent to that over A. Moreover, we show that almost all of the commonly encountered polymorphism properties are held equivalently on the A and the constructed digraph. As a consequence, the Algebraic CSP dichotomy conjecture as well as the conjectures characterizing CSPs solvable in logspace and in nondeterministic logspace are equivalent to their restriction to digraphs.