Monoidal Width-Versatile Pathway

• Width-Versatile Pathway is a unified algebraic framework that captures path, tree, and branch widths through monoidal decompositions in structured categories.
• The framework defines monoidal width using strict rules for atomic morphisms and composite operations, establishing precise correspondences with classical graph invariants.
• It offers algorithmic benefits by recasting dynamic programming over decompositions, with potential applications in quantum circuits, Petri nets, and diagram rewriting.

A width-versatile pathway refers to the unified framework provided by monoidal width—a measure of decomposition complexity for morphisms in monoidal categories—that simultaneously captures the classical graph-theoretic invariants of path width, tree width, and branch width. This pathway is realized through the formalism of monoidal decompositions, where the allowed shapes of decomposition trees (chains, trees, or arbitrary trees) yield precise correspondences to the respective classical width parameters. The resulting pathway is “width-versatile” in that adjusting the decomposition constraints interpolates smoothly among path-, tree-, and branch-width within a single algebraic calculus, avoiding ad-hoc or domain-specific definitions (Lavore et al., 2022).

1. Formal Definition of Monoidal Width

Let $\mathcal{C}$ be a strict monoidal category with monoidal product $\otimes$ and composition $\circ$. Fix a distinguished set of atomic morphisms $\Gamma \subseteq \mathrm{Mor}(\mathcal{C})$. A weight function $\omega$ assigns a non-negative integer to each $g \in \Gamma$ and is extended to composites such that $$\omega(g_1 \otimes g_2) = \omega(g_1) + \omega(g_2)$$ and $\omega(\otimes) = 0$.

A monoidal decomposition of $f: A \rightarrow B$ is a finite tree whose internal nodes are labeled by $\otimes$ or $\circ_X$ (composition along $X$), and whose leaves are labeled by atomic morphisms in $\Gamma$. Let $\mathrm{Dec}(f)$ denote the set of all such decomposition trees $d$. The decomposition must evaluate to $f$ in $\mathcal{C}$.

The width $\operatorname{wd}(d)$ of a decomposition $d\in\mathrm{Dec}(f)$ is defined inductively:

• If $d$ is a leaf labeled by $g\in\Gamma$, then $\operatorname{wd}(d)=\omega(g)$.
• If $d$’s root is $\otimes$ with children $d_1,d_2$, then $$\operatorname{wd}(d) = \max\{\operatorname{wd}(d_1), \operatorname{wd}(d_2)\}$$.
• If $d$’s root is $\circ_X$ with children $d_1:A\to X$ and $d_2:X\to B$, then $$\operatorname{wd}(d) = \max\{\operatorname{wd}(d_1),\; \omega(X),\; \operatorname{wd}(d_2)\}$$.

The monoidal width of $f$ is

$$\operatorname{mwd}(f) = \min_{d \in \mathrm{Dec}(f)} \operatorname{wd}(d).$$

Significant restricted variants are:

• Monoidal tree width $\operatorname{mTwd}(f)$: Only allow decomposition trees where each sequential composition $\circ_X$ features an atomic morphism on one side (“tree-shaped”).
• Monoidal path width $\operatorname{mPwd}(f)$: Forbid all $\otimes$-nodes so the decomposition is a sequential “chain.”

2. Specialization to Graphs and Recovery of Path Width

Specialization occurs in the symmetric monoidal category $\mathsf{Cospan}(\mathrm{UGraph})$, where objects are finite discrete vertex-sets $X$, and morphisms $X \to Y$ are cospans $$X \xrightarrow{\partial_1} G \xleftarrow{\partial_2} Y$$ with $G=(V,E)$ a finite undirected graph and $\partial_1$, $\partial_2$ labeling boundary vertices. Composition is via pushout; the monoidal product is disjoint union.

All cospans are atomic ($\Gamma$ is the class of all cospans). The weight $\omega$ of a cospan $X \rightarrow G \leftarrow Y$ is $\omega(\text{cospan}) = |V(G)|$.

A monoidal path decomposition of the cospan $g:\emptyset \rightarrow G \leftarrow \emptyset$ is a compositional chain $g = g_1 \circ g_2 \circ \dots \circ g_r$ with each $g_i$ an induced subgraph of $G$. It is proven that

$$\operatorname{mPwd}(g) = \operatorname{pathwidth}(G),$$

where the minimal decomposition width matches the maximal bag size in a classical path decomposition of $G$ [(Lavore et al., 2022), Props. 5.2, 5.3].

The bijection: From a classical path decomposition $(V_1, V_2, \ldots, V_r)$, form cospans with $X_i = V_i \cap V_{i+1}$ and $g_i: X_i \rightarrow G_i \rightarrow \emptyset$, with $G_i$ the subgraph induced by $V_i$. The maximum bag size matches the maximum boundary size, so the classical and monoidal definitions coincide.

3. Comparative Unification of Path-, Tree-, and Branch-Width

The unifying aspect of the width-versatile pathway is realized by interpreting graph width invariants as instances of monoidal width by restricting decomposition tree shapes in the cospan-of-graphs category:

Monoidal Variant	Decomposition Constraint	Classical Invariant	Theoretical Relation
$\operatorname{mPwd}$	Chains only (no $\otimes$)	Path width ($\operatorname{pathwidth}$)	Exact equality
$\operatorname{mTwd}$	Tree-shaped, one side atomic	Tree width ($\operatorname{treewidth}$)	$$\operatorname{treewidth}(G) \leq \operatorname{mTwd}(g) \leq 2 \operatorname{treewidth}(G)$$
$\operatorname{mwd}$	Fully general (arbitrary)	Branch width ($\operatorname{branchwidth}$)	$$1/2 \operatorname{branchwidth}(G) \leq \operatorname{mwd}(g) \leq \operatorname{branchwidth}(G) + 1$$

Path decompositions correspond to strictly “linear” strings of sequential compositions, tree decompositions allow for parallel (monoidal) $\otimes$ but with attachment restricted to a single “frontier” bag, and branch decompositions of graphs correspond to fully general expressions interleaving $\otimes$ and $\circ$.

4. Explicit Example: The 4-Cycle $C_4$

Let $G = C_4$ be the undirected 4-cycle with vertex set $V = \{v_1, v_2, v_3, v_4\}$. The classical pathwidth of $C_4$ is 2. Consider $g : \emptyset \rightarrow C_4 \leftarrow \emptyset$.

Classical path decomposition: One minimal decomposition uses bags $B_1 = \{v_1, v_2, v_4\}$ and $B_2 = \{v_2, v_3, v_4\}$, with maximal bag size 3. With the convention used in (Lavore et al., 2022), the width is 3 (no subtraction), matching the maximal bag size.

Monoidal path decomposition: Decompose $C_4$ into its atomic edges $e_1, e_2, e_3, e_4$ (each $\emptyset \rightarrow K_2 \leftarrow \emptyset$ with 2 vertices) and compose:

$$g = e_1 \circ_{ \{v_2\} } e_2 \circ_{ \{v_3\} } e_3 \circ_{ \{v_4\} } e_4$$

At each step, the boundary sets have size at most 3. Thus, $\operatorname{mPwd}(g) = 3$.

If one adopts the common graph theory normalization (subtracting 1), both methods yield width 2. This demonstrates that the monoidal pathway formalism accurately recovers the classical invariant.

5. Algebraic and Algorithmic Implications

The width-versatile pathway is uniform: a single abstract notion (monoidal width) recovers the three principal graph decompositional widths by varying only the decomposition tree constraint. Unlike ad-hoc or domain-specific invariants, this approach provides algebraic clarity: every graph decomposition corresponds to a syntactic expression constructed from $\otimes$ and $\circ$, and atomic generators (cospans). This language is amenable to generic diagram-rewriting algorithms and can be transferred to other domains (e.g., quantum circuits, Petri nets, string diagrams) by suitable choice of ambient monoidal category.

Algorithmically, dynamic-programming methods for decompositions (path, tree, or branch) can be recast as evaluation schemes for bounded-width monoidal expressions. This enables the development of generic compilation or optimization routines: given an arrow in a monoidal category and a bounded-width decomposition, one can produce efficient evaluation code within the same formalism.

6. Significance and Directions

Monoidal width, through its width-versatile pathway, delivers a compositional framework that unifies the three dominant measures of graph decomposition—path-, tree-, and branch-width—without ad-hoc constructions. The algebraic approach extends “width-based” reasoning beyond graphs and suggests algorithmic developments in any domain admitting a monoidal category interpretation. This framework is especially promising for cross-disciplinary applications where general decomposition-based techniques can be transferred or further developed (Lavore et al., 2022).