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Monotone Bounded Depth Formula Complexity of Graph Homomorphism Polynomials

Balagopal Komarath, Rohit Narayanan

TL;DR

This work introduces baggy elimination trees and the parameter $\lambda_Δ(H)$ to capture monotone bounded-depth formula complexity for graph homomorphism and colored isomorphism polynomials. It proves a tight characterization: for connected $H$ with more than two vertices, the Δ-product-depth monotone formula complexity of $\textsf{ColIso}_{H}$ is $\Theta(n^{\lambda_Δ(H)})$, with upper bounds induced by a tree-based recursive construction and matching lower bounds from parse-tree–tree conversions. The authors then demonstrate almost optimal separations between monotone circuits and monotone formulas at fixed product depths and establish a product-depth hierarchy showing separations between depth levels Δ and Δ−1. Overall, the paper links algebraic bounded-depth complexity to a graph-structural parameter, enabling depth-aware separations in monotone models and providing a unified framework that extends previous unbounded-depth characterizations.

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 $Δ$ and $Δ$ + 1 for all $Δ$ $\ge$ 1.

Monotone Bounded Depth Formula Complexity of Graph Homomorphism Polynomials

TL;DR

This work introduces baggy elimination trees and the parameter to capture monotone bounded-depth formula complexity for graph homomorphism and colored isomorphism polynomials. It proves a tight characterization: for connected with more than two vertices, the Δ-product-depth monotone formula complexity of is , with upper bounds induced by a tree-based recursive construction and matching lower bounds from parse-tree–tree conversions. The authors then demonstrate almost optimal separations between monotone circuits and monotone formulas at fixed product depths and establish a product-depth hierarchy showing separations between depth levels Δ and Δ−1. Overall, the paper links algebraic bounded-depth complexity to a graph-structural parameter, enabling depth-aware separations in monotone models and providing a unified framework that extends previous unbounded-depth characterizations.

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 and + 1 for all 1.

Paper Structure

This paper contains 6 sections, 4 theorems, 7 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Baggy Elimination Trees
  4. Characterizing Bounded Product Depth Monotone Formulas
  5. Separating Circuits from Formulas in Bounded Depth
  6. Product Depth Hierarchy for Monotone Formulas

Key Result

Theorem 1

For any connected graph $H$ on more than two vertices, the polynomial family $\mathsf{ColIso}_{H}$ has $\Delta$-product depth monotone formula complexity of $\Theta(n^{\lambda_\Delta(H)})$.

Figures (2)

  • Figure 1: Baggy elimination tree of product depth two for $P_7$
  • Figure 2: Parse tree to baggy elimination tree for $P_7$

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7: Baggy Elimination Tree
  • Example 1
  • Remark 1
  • Theorem 1
  • ...and 8 more