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Conservative Maltsev Constraint Satisfaction Problems

Manuel Bodirsky, Andrew Moorhead

TL;DR

This work addresses CSPs over finite structures with a conservative Maltsev polymorphism, showing they can be solved in the complexity class $\oplus L$ by a symmetric linear $\mathbb Z_2$-Datalog program. The authors develop a detailed structure theory for conservative minority algebras, including a tree-based representation (via leaf algebras) and a finite invariant-relations basis, then define the nested classes $\mathbf P^{n,k}$ and structures $\mathfrak P^{n,k}$ to build a pp-definable reduction scheme. Central to the approach are the Solve and Reduce procedures, which recursively reduce CSP$(\mathfrak P^{n,k})$ to the base case CSP$(\mathfrak P^{n,1})$ and solve a $\mathbb Z_2$-linear system; the process is implemented in symmetric linear $\mathbb Z_2$-Datalog, placing the problem in $\oplus L$. A companion paper completes the pp-construction from any conservative Maltsev structure to the basic class, reinforcing the universality of the approach. The results yield a full complexity-theoretic classification (up to logspace reductions) for conservative Maltsev CSPs and demonstrate a powerful descriptive-complexity bridge via $G$-Datalog.

Abstract

We show that for every finite structure B with a conservative Maltsev polymorphism, the constraint satisfaction problem for B can be solved by a symmetric linear Z2-Datalog program, and in particular is in the complexity class parity-L. The proof has two steps: we first present the result for a certain subclass whose polymorphism algebras are hereditarily subdirectly irreducible. We then show that every other structure in our class can be primitively positively constructed from one of the structures in the subclass. The second step requires different techniques and will be presented in a companion article.

Conservative Maltsev Constraint Satisfaction Problems

TL;DR

This work addresses CSPs over finite structures with a conservative Maltsev polymorphism, showing they can be solved in the complexity class by a symmetric linear -Datalog program. The authors develop a detailed structure theory for conservative minority algebras, including a tree-based representation (via leaf algebras) and a finite invariant-relations basis, then define the nested classes and structures to build a pp-definable reduction scheme. Central to the approach are the Solve and Reduce procedures, which recursively reduce CSP to the base case CSP and solve a -linear system; the process is implemented in symmetric linear -Datalog, placing the problem in . A companion paper completes the pp-construction from any conservative Maltsev structure to the basic class, reinforcing the universality of the approach. The results yield a full complexity-theoretic classification (up to logspace reductions) for conservative Maltsev CSPs and demonstrate a powerful descriptive-complexity bridge via -Datalog.

Abstract

We show that for every finite structure B with a conservative Maltsev polymorphism, the constraint satisfaction problem for B can be solved by a symmetric linear Z2-Datalog program, and in particular is in the complexity class parity-L. The proof has two steps: we first present the result for a certain subclass whose polymorphism algebras are hereditarily subdirectly irreducible. We then show that every other structure in our class can be primitively positively constructed from one of the structures in the subclass. The second step requires different techniques and will be presented in a companion article.
Paper Structure (32 sections, 50 theorems, 55 equations, 12 figures, 4 algorithms)

This paper contains 32 sections, 50 theorems, 55 equations, 12 figures, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Structures
  4. Algebras
  5. Congruences
  6. The structure of conservative minority algebras
  7. Block congruences
  8. Tree representations of conservative minority algebras
  9. Leaf algebras and tree representations
  10. Existence of representations
  11. Uniqueness of representations
  12. Subtrees
  13. Tree transformations
  14. Basic tree operations
  15. Invariant relations
  16. ...and 17 more sections

Key Result

Lemma 1

If $\mathfrak A$ and $\mathfrak B$ are structures with finite relational signatures, and every relation of ${\bf A}$ has a primitive positive definition in $\mathfrak B$, then there is a symmetric linear Datalog reduction reduction from $\mathop{\mathrm{CSP}}\nolimits(\mathfrak A)$ to $\mathop{\math

Figures (12)

  • Figure 1: A tree and a potato diagram.
  • Figure 2: (Full and maximal) Saplings.
  • Figure 3: The first few $\mathcal{P}_{n,k}$ and $\mathbf{P}^{n,k}$ for $n=3$.
  • Figure 4: A full subalgebra $\mathbf{P}^{6,5}_{(X,w)} \leq \mathbf{P}^{6,5}$.
  • Figure 5: Examples for $n=3, k=4$ of $\psi^{n,k}_i$ and $T^{n,k}_{i,j}$. A gray dot means that the element does not belong to $P^{n,k}_i$.
  • ...and 7 more figures

Theorems & Definitions (127)

  • Lemma 1: StarkeDiss
  • Theorem 2: GeigerBoKaKoRo
  • Lemma 3: Carbonnel Carbonnel16b (Lemma 1)
  • Lemma 4
  • proof
  • Proposition 5
  • proof
  • Corollary 6
  • proof
  • Corollary 7
  • ...and 117 more