Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing

2025-12-15

AI Summary

This work systematically characterizes the computational complexity of the homomorphism indistinguishability decision problem (HomInd(𝒻)) over graph classes, unifying relaxations of graph isomorphism such as quantum isomorphism and cospectrality. Methodologically, it integrates counting monadic second-order logic (CMSO₂), structural analysis via treewidth/pathwidth, polynomial identity testing (PIT), and algebraic automata theory. Key contributions are threefold: (1) It establishes, for the first time, a conditionally optimal randomized polynomial-time algorithm for HomInd(𝒻) on CMSO₂-definable graph classes of bounded treewidth; (2) It precisely classifies HomInd(𝒻) on CMSO₂-definable graph classes of bounded pathwidth as C=L-complete and provides a tight reduction to equivalence of multi-automata; (3) It shows that derandomizing the randomized algorithm is equivalent to PIT ∈ PTIME, thereby revealing the intrinsic algebraic hardness of the problem.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Quantum Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

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 $ extrm{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 $ extrm{HomInd}(mathcal{F})$, Seppelt (MFCS 2024) showed that $ extrm{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 $ extrm{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.

Problem

• Characterizes graph isomorphism relaxations via homomorphism indistinguishability over graph classes.
• Determines complexity of homomorphism indistinguishability for bounded treewidth and pathwidth graph classes.
• Connects homomorphism indistinguishability to multiplicity automata equivalence and polynomial identity testing.

Innovation

• Randomized polynomial time algorithm for bounded treewidth graph classes
• Conditional optimality based on polynomial identity testing assumptions
• Complexity reduction to C=L for bounded pathwidth graph classes