Monotone Bounded-Depth Complexity of Homomorphism Polynomials

2025-05-28

AI Summary

This paper investigates the computational complexity of homomorphism polynomials and colored subgraph polynomials for a fixed graph $ H $, under monotone algebraic circuits of bounded depth. We introduce a novel graph parameter $ mathrm{tw}_Delta(H) $, defined as the minimum width among all tree decompositions of depth at most $ Delta $, thereby establishing the first precise correspondence between structural depth constraints on graphs and circuit depth. Our approach combines restricted tree decompositions, homomorphism-based constructions, an $ H^dagger $-elimination transformation, and fine-grained reductions under the Exponential Time Hypothesis (ETH). We prove that any monotone product-depth-$ Delta $ circuit computing the homomorphism polynomial for $ H $ requires size $ Theta(n^{mathrm{tw}_Delta(H^dagger)+1}) $. This yields an optimal depth-hierarchy theorem: increasing circuit depth by one induces an exponential separation in computational power, thereby establishing a strict monotone depth hierarchy.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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 $ extit{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 $ extit{monotone}$ circuit complexity of the homomorphism polynomial for $H$ is $Theta(n^{mathrm{tw}(H)+1})$. In this paper, we characterize the power of monotone $ extit{bounded-depth}$ circuits for homomorphism and colorful subgraph polynomials. This leads us to discover a natural hierarchy of graph parameters $mathrm{tw}_Delta(H)$, for fixed $Delta in mathbb N$, which capture the width of tree-decompositions for $H$ when the underlying tree is required to have depth at most $Delta$. We prove that monotone circuits of product-depth $Delta$ computing the homomorphism polynomial for $H$ require size $Theta(n^{mathrm{tw}_Delta(H^{dagger})+1})$, 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.

Problem

• Characterize monotone bounded-depth circuits for homomorphism polynomials.
• Discover hierarchy of graph parameters for tree-decomposition width.
• Determine optimal depth hierarchy for monotone circuits via graph theory.

Innovation

• Monotone bounded-depth circuits analyze homomorphism polynomials
• Hierarchy of graph parameters captures tree-decomposition width
• Optimal depth hierarchy via graph-theoretic arguments