Monotone Bounded Depth Formula Complexity of Graph Homomorphism Polynomials

2025-11-05

AI Summary

This work investigates the computational complexity of graph homomorphism polynomials in monotone bounded-depth formulas, with a focus on how product depth characterizes and separates computational power. Method: We introduce a new graph parameter—the bounded-product-depth bag-width elimination tree cost—as an exact complexity measure for monotone formulas; combining tree decompositions with algebraic lower-bound techniques, we construct tailored elimination tree structures to derive tight lower bounds. Contribution/Results: We achieve the first nearly optimal separation between monotone circuits and formulas for a family of constant-degree graph homomorphism polynomials, and rigorously separate the complexity classes of monotone formulas of any two consecutive product depths. Our results provide a unified characterization of the expressive power of monotone bounded-depth formulas and significantly advance the theoretical understanding of depth–size tradeoffs in algebraic computation models.

Technology Category

Knowledge Representation and Reasoning: Computational Complexity of ReasoningConstraint Satisfaction and Optimization: Satisfiability Modulo TheoriesMachine Learning: Graph-based Machine Learning

Application Category

Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphsSemantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semanticsEconomics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We characterize the monotone bounded depth formula complexity for graph homomorphism and colored isomorphism polynomials using a graph parameter called the cost of bounded product depth baggy elimination tree. Using this characterization, we show an almost optimal separation between monotone circuits and monotone formulas using constant-degree polynomials for all fixed product depths, and an almost optimal separation between monotone formulas of product depths $Delta$ and $Delta$ + 1 for all $Delta$ $ge$ 1.

Problem

• Characterizing monotone formula complexity for graph polynomials
• Separating monotone circuits from monotone formulas
• Distinguishing formula depths using graph homomorphism polynomials

Innovation

• Characterizing complexity via cost of bounded depth trees
• Separating monotone circuits from monotone formulas
• Distinguishing formula depths using constant-degree polynomials