Structured Decompositions: A Categorical Approach to Generalize Combinatorial Invariants and Develop Compositional Algorithms

Core Concepts

Structured decompositions are category-theoretic data structures that generalize notions from graph theory, geometric group theory, and dynamical systems. They enable the generalization of combinatorial invariants like tree-width to new settings and the development of compositional algorithms that exploit the structural and algorithmic compositionality of their inputs.

Abstract

The paper introduces the concept of structured decompositions, which are category-theoretic data structures that generalize notions from graph theory, geometric group theory, and dynamical systems. Structured decompositions allow for the generalization of combinatorial invariants like tree-width to new settings beyond graphs.
The key ideas are:

Structured decompositions are diagrams in a fixed category, where objects are assigned to the vertices and edges of a graph. This generalizes the notion of graph decompositions.

The authors associate a notion of "width" to structured decompositions, not as a numeric invariant but as a category. This is done by starting with a subcategory Ω that picks out the "simple" or "atomic" objects in the category.

The authors show that many combinatorial invariants like tree-width, layered tree-width, and graph decomposition width can be captured by their framework of structured decompositions and (Ω,ω)-width.

For adhesive categories, the authors prove a connection between the (Ω,ω)-width of an object and the existence of a structured decomposition that exhibits how to construct that object as a colimit.

The authors also provide an algorithmic meta-theorem that, when instantiated in the category of graphs, yields compositional (albeit not FPT-time) algorithms for NP-hard problems like Maximum Bipartite Subgraph and Longest Path.

Overall, the paper introduces a powerful categorical framework for generalizing combinatorial invariants and developing compositional algorithms that exploit the structure of their inputs.

Stats

The tree-width of a graph 퐺 is defined as the minimum clique number of any chordal graph 퐻 that 퐺 can be mapped into.
The layered tree-width of a graph 퐺 is a tree-width variant that remains bounded on planar graphs.
Graph decomposition width is a recent notion related to coverings and fundamental groups of graphs.

Quotes

"Compositionality can be understood as the perspective that the meaning, structure or function of the whole is given by that of its constituent parts."
"Structured decompositions are special kinds of diagrams in some fixed category."
"The 'width' of a decomposition is not a natural number (as is customary in graph theory) but instead a category."

Key Insights Distilled From

Structured Decompositions: Structural and Algorithmic Compositionality

by Benjamin Mer... at arxiv.org 09-11-2024

https://arxiv.org/pdf/2207.06091.pdf

Deeper Inquiries

How can the ideas of structured decompositions be extended to other mathematical structures beyond graphs, such as higher-dimensional topological spaces or algebraic varieties?

The concept of structured decompositions, as introduced in the context of graphs, can indeed be extended to other mathematical structures, including higher-dimensional topological spaces and algebraic varieties. This extension can be achieved by leveraging the categorical framework that underpins structured decompositions.

Higher-Dimensional Topological Spaces: In topology, one can consider simplicial complexes or CW complexes as higher-dimensional analogs of graphs. A structured decomposition in this context could involve associating categorical objects to the vertices, edges, and higher-dimensional faces of a simplicial complex. By defining a functor that maps these complexes to a suitable category, one can create structured decompositions that capture the combinatorial and topological properties of the space. For instance, one could use the notion of a "nerve" of a cover to construct a structured decomposition that reflects the connectivity and higher-dimensional relationships within the space.

Algebraic Varieties: In algebraic geometry, structured decompositions can be applied to algebraic varieties by associating geometric objects (such as schemes or sheaves) to the points and subvarieties of a given variety. The functorial approach allows for the definition of decompositions that respect the algebraic structure, such as morphisms between varieties. This could lead to new insights into the geometric properties of varieties, such as their dimension and singularities, by analyzing the structured decompositions in terms of their categorical representations.

General Framework: The key to extending structured decompositions lies in the ability to define a suitable category that captures the essential features of the mathematical structure in question. By identifying a "spine" functor that selects objects of interest within the category, one can generalize the notion of width and develop compositional algorithms that exploit the structural properties of these higher-dimensional objects.

What are the limitations of the (Ω,ω)-width approach, and are there alternative ways to define width measures that could lead to even more general compositional algorithms?

The (Ω,ω)-width approach provides a powerful framework for defining width measures in a categorical context, but it does have certain limitations:

Dependence on Adhesiveness: The requirement that the category is adhesive can restrict the applicability of the (Ω,ω)-width approach. While many combinatorial objects reside in adhesive categories, there are important classes of structures that do not meet this criterion. This limitation may hinder the development of compositional algorithms for a broader range of mathematical objects.

Complexity of Width Measures: The definition of width as a category rather than a numeric invariant can complicate the interpretation and comparison of different width measures. This abstraction may obscure the practical implications of width in algorithmic contexts, making it challenging to derive concrete algorithmic results.

Alternative Definitions: To address these limitations, alternative approaches to defining width measures could be explored. For instance, one could consider numeric invariants that capture structural complexity while still allowing for categorical interpretations. This could involve defining width in terms of specific properties of morphisms or subobjects, potentially leading to more intuitive and computable measures.

General Compositional Algorithms: By exploring alternative definitions of width that do not rely on the adhesive property, one could develop more general compositional algorithms applicable to a wider variety of mathematical structures. This could involve integrating insights from other areas of mathematics, such as homotopy theory or category theory, to create a richer framework for understanding compositionality across different contexts.

Can the connections between structured decompositions and other category-theoretic notions like spined categories, undirected wiring diagrams, and monoidal width be further explored and exploited?

Yes, the connections between structured decompositions and other category-theoretic notions such as spined categories, undirected wiring diagrams, and monoidal width present rich avenues for exploration and exploitation:

Spined Categories: The relationship between structured decompositions and spined categories can be further investigated by examining how the spine functor can be generalized or adapted to different contexts. By understanding the interplay between the spine and the structured decomposition, researchers can potentially uncover new insights into the nature of compositionality and structural complexity in various mathematical settings.

Undirected Wiring Diagrams: The similarity between structured decompositions and undirected wiring diagrams suggests that there may be a deeper categorical framework that unifies these concepts. Exploring this connection could lead to the development of new tools and techniques for analyzing complex systems, particularly in areas such as network theory and systems biology, where wiring diagrams are prevalent.

Monoidal Width: The notion of monoidal width, which measures the complexity of morphisms in a monoidal category, can be related to structured decompositions by considering how the decomposition of objects can inform the understanding of morphisms. By establishing a bridge between these two concepts, one could develop new width measures that capture both object and morphism complexity, leading to more comprehensive compositional algorithms.

Interdisciplinary Applications: The exploration of these connections can also have interdisciplinary implications, as the ideas of structured decompositions and their related concepts can be applied to fields such as computer science, physics, and biology. By leveraging the categorical framework, researchers can develop new models and algorithms that reflect the underlying structural properties of complex systems across various domains.

In summary, the interplay between structured decompositions and other category-theoretic notions offers a fertile ground for further research, potentially leading to new insights and advancements in both theoretical and applied mathematics.