Monotone Bounded-Depth Complexity of Homomorphism Polynomials

TL;DR

This work develops a tight theory for the monotone bounded-depth complexity of homomorphism and colorful subgraph polynomials. By introducing the bounded-depth graph parameters $\mathrm{tw}_\Delta$ and its pruned version $\mathrm{ptw}_\Delta$, it shows that monotone product-depth $\Delta$ circuits computing $\mathsf{Hom}_{H,n}$ and $\mathsf{ColSub}_{H,n}$ must have size $\Theta\big(n^{\mathrm{ptw}_\Delta(H^{\dagger})+1}\big)$, with a matching bound for monotone ABPs via $\mathrm{ppw}_\Delta$. The paper also establishes a depth-hierarchy theorem, demonstrating separations between bounded-depth monotone circuits and deeper counterparts by constructing pattern graphs with optimal-size upper bounds at depth $\Delta+1$ but exponential lower bounds at depth $\Delta$. These results connect graph-structural parameters of $H$ with the inherent complexity of the corresponding algebraic counting polynomials, and extend prior monotone circuit characterizations (based on treewidth, treedepth, etc.) to bounded-depth models. The findings illuminate how bounded-height decompositions govern parallelism and size in monotone algebraic computation for pattern-count polynomials, offering precise, optimal separations and a robust hierarchy framework.

Abstract

For every fixed graph $H$, it is known that homomorphism counts from $H$ and colorful $H$-subgraph counts can be determined in $O(n^{t+1})$ time on $n$-vertex input graphs $G$, where $t$ is the treewidth of $H$. On the other hand, a running time of $n^{o(t / \log t)}$ would refute the exponential-time hypothesis. Komarath, Pandey and Rahul (Algorithmica, 2023) studied algebraic variants of these counting problems, i.e., homomorphism and subgraph $\textit{polynomials}$ for fixed graphs $H$. These polynomials are weighted sums over the objects counted above, where each object is weighted by the product of variables corresponding to edges contained in the object. As shown by Komarath et al., the $\textit{monotone}$ circuit complexity of the homomorphism polynomial for $H$ is $Θ(n^{\mathrm{tw}(H)+1})$. In this paper, we characterize the power of monotone $\textit{bounded-depth}$ circuits for homomorphism and colorful subgraph polynomials. This leads us to discover a natural hierarchy of graph parameters $\mathrm{tw}_Δ(H)$, for fixed $Δ\in \mathbb N$, which capture the width of tree-decompositions for $H$ when the underlying tree is required to have depth at most $Δ$. We prove that monotone circuits of product-depth $Δ$ computing the homomorphism polynomial for $H$ require size $Θ(n^{\mathrm{tw}_Δ(H^{\dagger})+1})$, where $H^{\dagger}$ is the graph obtained from $H$ by removing all degree-$1$ vertices. This allows us to derive an optimal depth hierarchy theorem for monotone bounded-depth circuits through graph-theoretic arguments.

Figures (1)

• Figure 1: Extracting a tree decomposition of height $2$ for $H^{\dagger}$ from a parse-tree of product-depth $2$ for a monomial of $\mathsf{ColSub}_{H,n}$. We have for all $i\in[6]$, $f_i := f(i) \in [n]$.

Theorems & Definitions (34)

• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1: Algebraic Circuits
• Definition 2.2: Skew Circuits
• Definition 2.3: Algebraic Branching Programs
• Definition 2.4: Monotone Circuits and ABPs
• Remark 2.5
• Definition 2.6: Parse Trees