Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing

Two graphs $G$ and $H$ are homomorphism indistinguishable over a graph class $\mathcal{F}$ if they admit the same number of homomorphisms from every graph $F \in \mathcal{F}$. Many graph isomorphism relaxations such as (quantum) isomorphism and cospectrality can be characterised as homomorphism indistinguishability over specific graph classes. Thereby, the problems $\textrm{HomInd}(\mathcal{F})$ of deciding homomorphism indistinguishability over $\mathcal{F}$ subsume diverse graph isomorphism relaxations whose complexities range from logspace to undecidable. Establishing the first general result on the complexity of $\textrm{HomInd}(\mathcal{F})$, Seppelt (MFCS 2024) showed that $\textrm{HomInd}(\mathcal{F})$ is in randomised polynomial time for every graph class $\mathcal{F}$ of bounded treewidth that can be defined in counting monadic second-order logic $\mathsf{CMSO}2$. We show that this algorithm is conditionally optimal, i.e. it cannot be derandomised unless polynomial identity testing is in $\mathsf{PTIME}$. For $\mathsf{CMSO}2$-definable graph classes $\mathcal{F}$ of bounded pathwidth, we improve the previous complexity upper bound for $\textrm{HomInd}(\mathcal{F})$ from $\mathsf{PTIME}$ to $\mathsf{C}=\mathsf{L}$ and show that this is tight. Secondarily, we establish a connection between homomorphism indistinguishability and multiplicity automata equivalence which allows us to pinpoint the complexity of the latter problem as $\mathsf{C}=\mathsf{L}$-complete.

💡 Research Summary

This paper investigates the computational complexity of the homomorphism indistinguishability problem (HomInd) for graphs, a unifying framework that captures many relaxations of graph isomorphism such as quantum isomorphism, cospectrality, and the k‑dimensional Weisfeiler–Leman test. Given a fixed class 𝔽 of graphs, HomInd(𝔽) asks whether two input graphs G and H have the same number of homomorphisms from every graph F∈𝔽. The authors focus on classes 𝔽 that are definable in counting monadic second‑order logic (CMSO₂) and have bounded treewidth or bounded pathwidth.

The first major contribution is a conditional optimality result for bounded‑treewidth CMSO₂ classes. Building on Seppelt’s recent randomized polynomial‑time (coRP) algorithm for HomInd(𝔽) when 𝔽 has bounded treewidth and is CMSO₂‑definable, the authors show that this randomness cannot be eliminated unless the polynomial identity testing problem (PIT) lies in deterministic polynomial time. To achieve this, they construct a log‑space many‑one reduction chain: HomInd(𝔽₀) → multiplicity tree automaton (MTA) equivalence → PIT → HomInd(𝔽₀), where 𝔽₀ is a specific CMSO₂‑definable class of bounded treewidth. Since PIT is a canonical problem whose deterministic polynomial‑time solvability would have far‑reaching consequences, the reduction demonstrates that any deterministic polynomial‑time algorithm for HomInd(𝔽₀) would imply PIT∈P. Consequently, the existing randomized algorithm is conditionally optimal.

The second contribution refines the upper bound for bounded‑pathwidth CMSO₂ classes. Previously, HomInd(𝔽) for such classes was known to be in PTIME. The authors improve this to the class C = L (also known as DET), which consists of languages decidable by testing whether a GapL function equals zero. The key insight is a reduction from HomInd(𝔽₁) (for any bounded‑pathwidth CMSO₂‑definable class 𝔽₁) to equivalence of multiplicity word automata (MWA). Using Tzeng’s classic result that MWA equivalence is C = L‑complete, they obtain: HomInd(𝔽₁) ≤_L MWA‑equivalence ≤_L HomInd(𝔽₁), establishing that HomInd(𝔽₁) is C = L‑complete. This places the problem firmly within log‑space, and because C = L ⊆ NC², the decision can be parallelized efficiently (poly‑processor, O(log² n) time).

A third, auxiliary result proves that MWA equivalence itself is C = L‑complete under log‑space many‑one reductions. This complements earlier work showing that MTA equivalence is log‑space reducible to PIT, thereby linking three distinct domains: graph homomorphism counting, algebraic circuit identity testing, and weighted automata theory.

The paper also surveys related work. Grohe showed that HomInd over all graphs of treewidth ≤ k is P‑complete, while Raßmann, Schindling, and Schweitzer proved that HomInd over graphs of treedepth ≤ k lies in L. However, these results apply only to specific families and do not address the broader CMSO₂‑definable classes considered here. The present work thus fills a gap by providing a uniform complexity classification for all CMSO₂‑definable classes of bounded treewidth or pathwidth.

In summary, the authors achieve three main outcomes:

1. They prove that for some CMSO₂‑definable bounded‑treewidth class 𝔽₀, HomInd(𝔽₀) is log‑space many‑one equivalent to PIT and MTA equivalence, making the known randomized algorithm conditionally optimal.
2. They show that for any CMSO₂‑definable bounded‑pathwidth class 𝔽₁, HomInd(𝔽₁) is C = L‑complete, yielding a deterministic log‑space algorithm and a tight complexity bound.
3. They establish C = L‑completeness of MWA equivalence, thereby connecting weighted automata equivalence to classic linear‑algebraic problems.

These results deepen the theoretical understanding of graph isomorphism relaxations, clarify the role of randomness in their algorithms, and open avenues for further exploration of homomorphism indistinguishability across richer logical fragments and graph families.