Algorithmic and Extremal Obstructions Through the Language of Cohomology

TL;DR

This paper introduces a cohomological framework using presheaves and ch cohomology to analyze algorithmic problems, revealing obstructions to solutions and unifying classical results with modern topological methods.

Contribution

It develops a systematic, sheaf-theoretic approach to understanding algorithmic obstructions, connecting graph problems with cohomology and extending to Abelian presheaves.

Findings

Rephrases classical graph theorems in cohomological terms

Identifies hidden cycles and local solution inflation as obstructions

Provides a new perspective on the failure of local-to-global solutions

Abstract

We model problems as presheaves that assign sets of certificates to input instances, and we show how to use presheaf \v{C}ech 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 K\"onig'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.