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

The border, or the approximative, model of algebraic computation (VP) is quite popular due to the Geometric Complexity Theory (GCT) approach to P≠NP conjecture, and its complex analytic origins. On the flip side, the definition of the border is inherently existential in the field constants that the model employs. In particular, a poly-size border circuit C(ε, x) cannot be compactly presented in reality, as the limit parameter ε may require exponential precision. In this work we resolve this issue by giving a constructive, or a presentable, version of border circuits and state its applications. We make border presentable by restricting the circuit C to use only those constants, in the function field Fq(ε), that it can generate by the ring operations on {ε}∪Fq, and their division, within poly-size circuit. This model is more expressive than VP as it affords exponential-degree in ε; and analogous to the usual border, we define new border classes called VPε and VNPε. We prove that both these (now called presentable border) classes lie in VNP. Such a ’debordering’ result is not known for the classical border classes VP and respectively for VNP. We pose VPε=VP as a new conjecture to study the border. The heart of our technique is a newly formulated exponential interpolation over a finite field, to bound the Boolean complexity of the coefficients before deducing the algebraic complexity. It attacks two factorization problems which were open before. We make progress on (Conj.8.3 in Bürgisser 2000, FOCS 2001) and solve (Conj.2.1 in Bürgisser 2000; Chou,Kumar,Solomon CCC 2018) over all finite fields: 1. Each poly-degree irreducible factor, with multiplicity coprime to field characteristic, of a poly-size circuit (of possibly exponential-degree), is in VNP. 2. For all finite fields, and all factors, VNP is closed under factoring. Consequently, factors of VP are always in VNP. The prime characteristic cases were open before due to the inseparability obstruction (i.e. ‍when the multiplicity is not coprime to q).