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Digraphs modulo primitive positive constructability

Florian Starke

TL;DR

This thesis develops a comprehensive algebraic framework for studying the pp-constructability order on finite digraphs and related structures. It interrelates pp-constructability, minor conditions, polymorphism clones, and free structures to characterize when one structure pp-constructs another, and how this impacts CSP complexity via log-space reductions. Central contributions include a detailed description of the pp-constructability poset, the Polymorphism Preservation Theorem, and the free-structure approach, plus applications to directed cycles, smooth digraphs, and related classes. The work advances understanding of CSP dichotomy phenomena for digraphs and provides lattice-theoretic insights into how pp-constructability interacts with polymorphism properties and Datalog fragments, with several open questions and directions for further research.

Abstract

This is my dissertation about digraphs ordered by pp-constructability. We study in particular smooth digraphs, i.e., digraphs without sources or sinks, tournaments and semicomplete digraphs, orientations of paths and cycles, digraphs with at most four vertices, and orientations of trees.

Digraphs modulo primitive positive constructability

TL;DR

This thesis develops a comprehensive algebraic framework for studying the pp-constructability order on finite digraphs and related structures. It interrelates pp-constructability, minor conditions, polymorphism clones, and free structures to characterize when one structure pp-constructs another, and how this impacts CSP complexity via log-space reductions. Central contributions include a detailed description of the pp-constructability poset, the Polymorphism Preservation Theorem, and the free-structure approach, plus applications to directed cycles, smooth digraphs, and related classes. The work advances understanding of CSP dichotomy phenomena for digraphs and provides lattice-theoretic insights into how pp-constructability interacts with polymorphism properties and Datalog fragments, with several open questions and directions for further research.

Abstract

This is my dissertation about digraphs ordered by pp-constructability. We study in particular smooth digraphs, i.e., digraphs without sources or sinks, tournaments and semicomplete digraphs, orientations of paths and cycles, digraphs with at most four vertices, and orientations of trees.
Paper Structure (6 sections, 11 theorems, 35 equations, 3 figures)

This paper contains 6 sections, 11 theorems, 35 equations, 3 figures.

Table of Contents

  1. Preliminaries
  2. The pp-constructability Poset
  3. Minor Conditions
  4. The Polymorphism Preservation Theorem
  5. How I Learned to Love Free Structures
  6. Constraint Satisfaction Problems

Key Result

Corollary 1

Let $a$ be in $\mathop{\mathrm{\mathbb N}}\nolimits^+$ and let $\mathop{\mathrm{\mathbb B}}\nolimits$ be a finite structure with finite relational signature. Then where $\mathop{\mathrm{\mathbb C}}\nolimits_a$ is the directed cycle of length $a$ and $\Sigma_p$ is the $p$-cyclic loop condition.

Figures (3)

  • Figure 1: Complexity classes from the Larose-Tesson Conjecture ordered by inclusion (left) and structures whose CSPs are complete for the respective complexity class (right).
  • Figure 2: The structure $\mathop{\mathrm{\mathbb C}}\nolimits_6$ pp-constructs $\mathop{\mathrm{\mathbb C}}\nolimits_3$.
  • Figure 3: Diagram of $\lambda(f)=h_b\circ f\circ h_a$.

Theorems & Definitions (32)

  • Conjecture 1: LaroseTesson
  • Corollary 1: \ref{['cor:PPvsLoopCaBlockerForSa']}
  • Corollary 2: \ref{['cor:cyclesSquarefree']}
  • Theorem 1: \ref{['thm:lowerCoversOfSubmaximalCycles']}
  • Corollary 3: \ref{['cor:NLDichotomyHoldForPaths']}
  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Lemma 1.4
  • proof
  • ...and 22 more