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Algorithmic and Extremal Obstructions Through the Language of Cohomology

Anny Beatriz Azevedo, Benjamin Merlin Bumpus, Matteo Capucci, James Fairbanks, Daniel Rosiak

TL;DR

This work proposes a presheaf-based, cohomological framework to study obstructions in algorithmic problems, with the zeroth Čech cohomology $H^0$ capturing failures of local solutions to patch into a global solution and obstructions to compositionality. By modeling problems such as VertexCover, CycleCover, and OddCycleTransversal as presheaves over subgraph categories, the authors show how to compute obstructions via $H^0$ and how sheafification $F^+$ relates to the existence of global solutions. They develop general machinery (plus construction, sheafification) and demonstrate that, for separated presheaves, $H^0(-,F)$ equals the cokernel of $F o F^+$, enabling explicit, computable obstructions such as size-based barriers in VertexCover and emergent cycles in CycleCover. A model-collecting functor $\mathfrak{M}$ is introduced to produce flasque presheaves, yielding cohomological formulations of solvability and connecting to König-type results; this provides a principled route to study not only obstructions to existence but also obstructions to composition in algorithmic settings. The framework offers a promising bridge between graph theory, topology, and category theory, with potential extensions to Abelian presheaves and higher cohomology for deeper obstruction theory and extremal results.

Abstract

We model problems as presheaves that assign sets of certificates to input instances, and we show how to use presheaf Čech cohomology to capture the precise ways in which local solutions fail to patch into global ones. Applied to problems like Vertex Cover, Cycle Cover, and Odd Cycle Transversal, our framework exposes emergent phenomena such as hidden cycles or the inflation of small, local solutions. This approach not only rephrases classical results like König's Theorem in cohomological terms, but also reveals how to systematically account for failures of compositionality. Although our main focus is on presheaves of sets, the methods generalize naturally to Abelian presheaves, suggesting a rich interplay between graph theory, cohomology, and complexity. This work represents a first step toward a systematic, sheaf-theoretic theory of algorithmic structure and related obstructions.

Algorithmic and Extremal Obstructions Through the Language of Cohomology

TL;DR

This work proposes a presheaf-based, cohomological framework to study obstructions in algorithmic problems, with the zeroth Čech cohomology capturing failures of local solutions to patch into a global solution and obstructions to compositionality. By modeling problems such as VertexCover, CycleCover, and OddCycleTransversal as presheaves over subgraph categories, the authors show how to compute obstructions via and how sheafification relates to the existence of global solutions. They develop general machinery (plus construction, sheafification) and demonstrate that, for separated presheaves, equals the cokernel of , enabling explicit, computable obstructions such as size-based barriers in VertexCover and emergent cycles in CycleCover. A model-collecting functor is introduced to produce flasque presheaves, yielding cohomological formulations of solvability and connecting to König-type results; this provides a principled route to study not only obstructions to existence but also obstructions to composition in algorithmic settings. The framework offers a promising bridge between graph theory, topology, and category theory, with potential extensions to Abelian presheaves and higher cohomology for deeper obstruction theory and extremal results.

Abstract

We model problems as presheaves that assign sets of certificates to input instances, and we show how to use presheaf Čech cohomology to capture the precise ways in which local solutions fail to patch into global ones. Applied to problems like Vertex Cover, Cycle Cover, and Odd Cycle Transversal, our framework exposes emergent phenomena such as hidden cycles or the inflation of small, local solutions. This approach not only rephrases classical results like König's Theorem in cohomological terms, but also reveals how to systematically account for failures of compositionality. Although our main focus is on presheaves of sets, the methods generalize naturally to Abelian presheaves, suggesting a rich interplay between graph theory, cohomology, and complexity. This work represents a first step toward a systematic, sheaf-theoretic theory of algorithmic structure and related obstructions.
Paper Structure (14 sections, 75 equations, 1 figure)

This paper contains 14 sections, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Motivation: Naive Dynamic Programming
  3. Presheaves and Čech Cohomology: VertexCover as a Case Study
  4. Warmup: VertexCover as a Presheaf
  5. Some Background on Cohomology: Obstructions to being a sheaf
  6. The Zeroth Čech Cohomology of Computational Problems
  7. Developing Some Technical Machinery
  8. More computational problems as examples
  9. Categorical Generalities to Construct Presheaves from Monotone Graph Properties
  10. More Examples: CycleCover and OddCycleTransversal
  11. The Compositionality of $H^0$
  12. Model collecting
  13. Proof of Theorem \ref{['thm: model collecting']}
  14. Discussion & Future Work

Figures (1)

  • Figure 1: The presheaf with domain $\mathop{\mathrm{\mathsf{Sub}}}\nolimits(K_2)$ associated to the problem of deciding if a graph admits a vertex cover of cardinality at most $1$.