Monotone Bounded-Depth Complexity of Homomorphism Polynomials

TL;DR

This paper explores the complexity of homomorphism polynomials for fixed graphs, establishing a hierarchy based on bounded-depth monotone circuits and introducing new graph parameters related to tree-decomposition width.

Contribution

It characterizes the power of monotone bounded-depth circuits for homomorphism and subgraph polynomials, introducing the hierarchy of parameters _0(H) and proving size bounds.

Findings

Monotone circuits of product-depth  compute homomorphism polynomials with size (H^{})+1.

Introduces the hierarchy of graph parameters _0(H) capturing bounded-depth circuit complexity.

Establishes an optimal depth hierarchy theorem for monotone bounded-depth circuits.

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 $\Theta(n^{\mathrm{tw}(H)+1})$. 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 $\mathrm{tw}_\Delta(H)$, 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.