Algorithmic and Extremal Obstructions Through the Language of Cohomology
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arXiv
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| Format: | Preprint |
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2024
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| author | Azevedo, Anny Beatriz Bumpus, Benjamin Merlin Capucci, Matteo Fairbanks, James Rosiak, Daniel |
| author_facet | Azevedo, Anny Beatriz Bumpus, Benjamin Merlin Capucci, Matteo Fairbanks, James Rosiak, Daniel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03488 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Algorithmic and Extremal Obstructions Through the Language of Cohomology Azevedo, Anny Beatriz Bumpus, Benjamin Merlin Capucci, Matteo Fairbanks, James Rosiak, Daniel Commutative Algebra Category Theory 18G90 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. |
| title | Algorithmic and Extremal Obstructions Through the Language of Cohomology |
| topic | Commutative Algebra Category Theory 18G90 |
| url | https://arxiv.org/abs/2407.03488 |