Hardness of Network Satisfaction for Relation Algebras with Normal Representations

TL;DR

This paper investigates the computational complexity of network satisfaction problems in relation algebras with normal representations, establishing NP-hardness under specific structural conditions.

Contribution

It identifies new conditions under which the network satisfaction problem for finite relation algebras is NP-hard, expanding understanding of their computational complexity.

Findings

NP-hardness when the representation contains a finite equivalence relation

NP-hardness for domain size at least three with certain forbidden triples

Application of conditions to small relation algebras

Abstract

We study the computational complexity of the general network satisfaction problem for a finite relation algebra $A$ with a normal representation $B$. If $B$ contains a non-trivial equivalence relation with a finite number of equivalence classes, then the network satisfaction problem for $A$ is NP-hard. As a second result, we prove hardness if $B$ has domain size at least three and contains no non-trivial equivalence relations but a symmetric atom $a$ with a forbidden triple $(a,a,a)$, that is, $a \not\leq a \circ a$. We illustrate how to apply our conditions on two small relation algebras.