Monotone Bounded Depth Formula Complexity of Graph Homomorphism Polynomials

TL;DR

This paper characterizes the monotone bounded depth formula complexity for graph homomorphism polynomials, revealing near-optimal separations between different classes of monotone formulas and circuits using a new graph parameter.

Contribution

Introduces a graph parameter called the cost of bounded product depth baggy elimination tree to characterize monotone bounded depth formula complexity for graph homomorphism polynomials.

Findings

Established almost optimal separation between monotone circuits and formulas.

Proved near-optimal separation between formulas of product depths Δ and Δ+1.

Provided a new graph parameter for complexity characterization.

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.