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Flexible constraint satisfiability and a problem in semigroup theory

Marcel Jackson

TL;DR

This work investigates flexible notions of constraint satisfaction and their connections to model-theoretic Horn classes, culminating in explicit hardness results for finite-membership problems in algebraic varieties. By constructing a chain of reductions from robust SAT variants to CSP and then to pseudovariety membership, the paper identifies the six-element Brandt monoid \mathbf{B}_2^1 as a keystone example and proves NP-hardness for a broad interval of finite algebras, including a 3-element algebra and a 4-element groupoid. It also extends these methods to other algebraic signatures, showing NP-hardness or NP-membership results in inverse semigroups and semirings, thereby mapping the minimal sizes and signatures that yield intractable membership problems. The results illuminate the deep links between robust CSP, universal Horn classes, and the complexity landscape of finite algebraic varieties, with implications for both theory and potential applications in algebraic CSP frameworks.

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.

Flexible constraint satisfiability and a problem in semigroup theory

TL;DR

This work investigates flexible notions of constraint satisfaction and their connections to model-theoretic Horn classes, culminating in explicit hardness results for finite-membership problems in algebraic varieties. By constructing a chain of reductions from robust SAT variants to CSP and then to pseudovariety membership, the paper identifies the six-element Brandt monoid \mathbf{B}_2^1 as a keystone example and proves NP-hardness for a broad interval of finite algebras, including a 3-element algebra and a 4-element groupoid. It also extends these methods to other algebraic signatures, showing NP-hardness or NP-membership results in inverse semigroups and semirings, thereby mapping the minimal sizes and signatures that yield intractable membership problems. The results illuminate the deep links between robust CSP, universal Horn classes, and the complexity landscape of finite algebraic varieties, with implications for both theory and potential applications in algebraic CSP frameworks.

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 -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 -element algebra with this property, and solve a widely stated problem by showing that the -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.

Paper Structure

This paper contains 14 sections, 21 theorems, 28 equations, 3 figures, 2 tables.

Table of Contents

  1. Preliminaries: constraints, satisfaction and separation
  2. Constraint Satisfaction Problems.
  3. Separation and unfrozen relations
  4. Robust satisfaction
  5. Flexible satisfaction
  6. Robust NAE3SAT
  7. Edge and vertex robust G3C
  8. Some comments.
  9. $2$-robust positive 1-in-3SAT and separability.
  10. Applications to variety membership
  11. Variety membership for the semigroup ${\bf B}_2^1$
  12. Variety membership for other algebraic signatures
  13. Variety membership for some small algebraic structures
  14. ${\bf B}_2^1$ in other signatures

Key Result

Theorem 2.1

Let $\mathbb{A}$ be a finite relational structure and $\mathbb{B}$ a relational structure of the same finite signature $\mathcal{R}$. Let $\{R_1,\dots,R_k\}$ be a subset of $\mathcal{R}\cup\{{=}\}$. The following are equivalent.

Figures (3)

  • Figure 1: Three finite graphs generating universal Horn classes with various stages of unfrozenness relative to homomorphisms into $\mathbb{K}_3$; see Example \ref{['eg:K3']}.
  • Figure 2: A schematic of $\mathbb{G}_{I}$
  • Figure 3: Three local configurations forcing the number of colours on $\{i_1,i_2,i_1',i_2'\}$ in any valid $3$-colouring (other edges and vertices may also be present but the $\exists$ symbol means that such a vertex does appear). In the first, exactly two colours can appear in $\{i_1,i_2,i_1',i_2'\}$. In the second and third, exactly $3$ colours appear in $\{i_1,i_2,i_1',i_2'\}$t. In the third configuration, we allow $i_2'$ to coincide with one of $i_1$ or $i_2$.

Theorems & Definitions (52)

  • Remark 1
  • Theorem 2.1
  • proof
  • Example 2.2
  • proof
  • Example 2.3
  • proof
  • Remark 2.4
  • Lemma 3.1
  • proof
  • ...and 42 more