Does this skill require Julia knowledge?

While it is based on StructuredDecompositions.jl, the skill helps Claude understand and implement these concepts even if you are just beginning to work with AlgebraicJulia.

What is the primary purpose of the Structured Decompositions skill?

It provides a framework for using sheaves on tree decompositions to solve mathematical and algorithmic problems more efficiently than standard graph-based methods.

How does it improve computational efficiency?

By utilizing Fixed-Parameter Tractable (FPT) algorithms, it reduces the runtime of complex problems to a linear function of input size, provided the 'width' of the decomposition remains small.

What is the 𝐃 Functor in this context?

The 𝐃 Functor is a mathematical tool that lifts a decision problem (like k-coloring) from a general category into a decomposition space where it can be solved locally.

Structured Decompositions

Name: Structured Decompositions
Author: plurigrid

byplurigrid

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データサイエンスとML

Applies category theory and sheaves on tree decompositions to solve complex combinatorial problems with Fixed-Parameter Tractable (FPT) algorithms.

概要

The Structured Decompositions skill integrates the principles of AlgebraicJulia and StructuredDecompositions.jl into your workflow. It allows Claude to model decision problems as functors and lift them into decomposition spaces using the 𝐃 Functor. By leveraging sheaves on tree decompositions, this skill enables the efficient solution of NP-hard problems, such as k-coloring, by optimizing for tree-width. It is an essential tool for researchers and developers working with higher-order graph theory, category theory-based modeling, and computational systems that require topological analysis.

主な機能

Lift decision problems to decomposition space via the 𝐃 Functor
Construct functors from shape graphs to target categories like Graph or FinSet
Implement sheaf-based solvers for tree-shaped decomposition structures
Optimize algorithmic runtime using Fixed-Parameter Tractable (FPT) techniques
2 GitHub stars
Automate extraction of bags, adhesions, and adhesion spans from decompositions

ユースケース

Modeling complex system architectures using algebraic category theory
Solving NP-hard graph problems on large-scale structured networks
Developing efficient algorithms for datasets with low-width tree decompositions

概要

主な機能

Lift decision problems to decomposition space via the 𝐃 Functor
Construct functors from shape graphs to target categories like Graph or FinSet
Implement sheaf-based solvers for tree-shaped decomposition structures
Optimize algorithmic runtime using Fixed-Parameter Tractable (FPT) techniques
2 GitHub stars
Automate extraction of bags, adhesions, and adhesion spans from decompositions

ユースケース

Modeling complex system architectures using algebraic category theory
Solving NP-hard graph problems on large-scale structured networks
Developing efficient algorithms for datasets with low-width tree decompositions