[Paper Review] Monoidal Width: Unifying Tree Width, Path Width and Branch Width

Elena Di Lavore, Paweł Sobociński|arXiv (Cornell University)|Feb 15, 2022

Logic, programming, and type systems4 citations

TL;DR

This paper introduces monoidal width as a unified algebraic framework for measuring the complexity of morphisms in monoidal categories, generalizing graph width measures like tree width, path width, and branch width. By instantiating monoidal width in the category of cospans of graphs, the authors prove that it precisely captures these classical graph widths, establishing tight bounds: 1/2 ⋅ bwd(G) ≤ mwd(g) ≤ bwd(G) + 1 for branch width, and analogous equalities for tree and path width.

ABSTRACT

We introduce monoidal width as a measure of the difficulty of decomposing morphisms in monoidal categories. For graphs, we show that monoidal width and two variations capture existing notions, namely branch width, tree width and path width. We propose that monoidal width: (i) is a promising concept that, while capturing known measures, can similarly be instantiated in other settings, avoiding the need for ad-hoc domain-specific definitions and (ii) comes with a general, formal algebraic notion of decomposition using the language of monoidal categories.

Motivation & Objective

• To unify existing graph width measures—tree width, path width, and branch width—under a single formal framework.
• To address the lack of a general, algebraic decomposition theory for computational artifacts by introducing monoidal width as a formal, executable notion of decomposition.
• To demonstrate that monoidal width generalizes known graph width measures when instantiated in the category of cospans of graphs.
• To establish formal bounds between monoidal width and classical graph widths, showing their equivalence under precise conditions.
• To lay the foundation for extending width measures to other structures, such as directed graphs, hypergraphs, and relational systems, via category-theoretic generalization.

Proposed method

• Define monoidal width as the minimal cost of decomposing a morphism using sequential and monoidal (parallel) composition in a monoidal category.
• Introduce restricted variants: monoidal tree width and monoidal path width, corresponding to tree-shaped and path-shaped decomposition trees.
• Construct mappings between recursive graph decompositions (e.g., recursive branch, tree, or path decompositions) and monoidal decompositions in the category of cospans of graphs.
• Use category-theoretic tools, including graph epimorphisms and subgraph inclusions, to preserve decomposition constraints and widths across mappings.
• Prove that the width of a monoidal decomposition matches the classical width of the corresponding graph decomposition via recursive structural induction.
• Establish tight quantitative bounds: mwd(g) ∈ [1/2 ⋅ bwd(G), bwd(G) + 1], with equality for tree and path width.

Experimental results

Research questions

• RQ1 Can a single, general notion of width be defined for morphisms in monoidal categories that subsumes existing graph width measures?
• RQ2 How do monoidal width and its variants relate to classical graph widths like branch width, tree width, and path width?
• RQ3 Can the mappings between recursive graph decompositions and monoidal decompositions be formalized and proven to preserve width?
• RQ4 What are the quantitative bounds between monoidal width and classical graph widths?
• RQ5 Can monoidal width be extended to other structures such as directed graphs, hypergraphs, or relational systems by choosing appropriate monoidal categories?

Key findings

• Monoidal width in the category of cospans of graphs recovers classical graph width measures: monoidal tree width equals tree width, monoidal path width equals path width, and monoidal width is bounded by branch width.
• For any graph G, the monoidal width of its corresponding cospan morphism g satisfies 1/2 ⋅ bwd(G) ≤ mwd(g) ≤ bwd(G) + 1.
• Monoidal tree width of a morphism g is equal to the tree width of the underlying graph: mtwd(g) = twd(G).
• Monoidal path width of a morphism g is equal to the path width of the underlying graph: mpwd(g) = pwd(G).
• The mapping from recursive branch decompositions to monoidal decompositions preserves width up to an additive +1, and vice versa up to a multiplicative factor of 2.
• The results establish that monoidal width is not only a unifying concept but also a formally precise and executable generalization of graph width, with potential for algorithmic applications via generalizations of Courcelle’s theorem.