What are structured decompositions in this context?

Structured decompositions are a category-theoretic generalization of tree decompositions. They allow complex global problems to be broken down into local 'bags' connected by 'adhesions' (shared boundaries).

What is the benefit of using sheaves for computation?

Using sheaves allows the system to verify that local solutions are consistent across overlaps. If local solutions satisfy the sheaf condition, they can be 'glued' together to form a valid global solution automatically.

How does this skill help with FPT algorithms?

It provides the mathematical framework to implement Fixed-Parameter Tractable algorithms, ensuring that if the decomposition width is small, the problem can be solved in linear time relative to the total data size.

Does it support standard Julia graphs?

Yes, it integrates with Graphs.jl, MetaGraphs.jl, and Catlab.jl, allowing you to perform decompositions on standard networks, hypergraphs, and attributed C-sets.

Structured Decompositions & Sheaves

Name: Structured Decompositions & Sheaves
Author: plurigrid

byplurigrid

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Data Science & ML

Implements sheaves on tree decompositions to solve combinatorial problems in linear time using Fixed-Parameter Tractable (FPT) algorithms.

This skill leverages the StructuredDecompositions.jl ecosystem to bridge category theory and computer science, focusing on compositional algorithms for compositional data. By representing computational problems as sheaves over tree decompositions, it enables the efficient resolution of NP-hard problems on structured data like graphs, hypergraphs, and databases in linear time (relative to input size). It provides robust bidirectional navigation of decomposition components—bags and adhesions—and integrates with the AlgebraicJulia suite for advanced algebraic data manipulation and FPT complexity management.

Key Features

01Tree decomposition generation and management for ACSet-based structures

028 GitHub stars

03S-expression serialization for inspection and roundtrip persistence

04Sheaf-based algorithm implementation for linear-time decision problems

05Bidirectional navigation of bags and adhesions via Specter integration

06Complexity optimization for bounded width classes (FPT patterns)

Use Cases

01Solving graph coloring or homomorphism problems on large, structured networks

02Decomposing metabolic or molecular graphs for high-performance scientific simulations

03Applying category-theoretic lenses to navigate complex hierarchical database schemas

What are Skills?·How to Install

Install with 🐟 Skill.Fish

npx skillfish add plurigrid/asi structured-decomp

For use in Claude.ai and ChatGPT

Download Skill