Flexible constraint satisfiability and a problem in semigroup theory

TL;DR

This paper explores flexible notions of constraint satisfaction, establishing NP-completeness results for certain problems and providing small algebraic examples with NP-hard variety membership, linking model theory and algebra.

Contribution

It introduces new NP-hardness results for algebraic variety membership problems and provides minimal examples, including a 3-element algebra and the 6-element Brandt monoid.

Findings

NP-completeness of 2-robust monotone 1-in-3 3SAT

Smallest algebra with NP-hard variety membership is 3-element

Smallest semigroup with NP-hard variety membership is 6-element

Abstract

We examine some flexible notions of constraint satisfaction, observing some relationships between model theoretic notions of universal Horn class membership and robust satisfiability. We show the \texttt{NP}-completeness of $2$-robust monotone 1-in-3 3SAT in order to give very small examples of finite algebras with \texttt{NP}-hard variety membership problem. In particular we give a $3$-element algebra with this property, and solve a widely stated problem by showing that the $6$-element Brandt monoid has \texttt{NP}-hard variety membership problem. These are the smallest possible sizes for a general algebra and a semigroup to exhibit \texttt{NP}-hardness for the membership problem of finite algebras in finitely generated varieties.