Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing

Marek Černý; Tim Seppelt

Paper

Homomorphism Indistinguishability, Multiplicity Automata Equivalence, and Polynomial Identity Testing

Abstract

Two graphs

and

are homomorphism indistinguishable over a graph class

if they admit the same number of homomorphisms from every graph

. Many graph isomorphism relaxations such as (quantum) isomorphism and cospectrality can be characterised as homomorphism indistinguishability over specific graph classes. Thereby, the problems

of deciding homomorphism indistinguishability over

subsume diverse graph isomorphism relaxations whose complexities range from logspace to undecidable. Establishing the first general result on the complexity of

, Seppelt (MFCS 2024) showed that

is in randomised polynomial time for every graph class

of bounded treewidth that can be defined in counting monadic second-order logic

. We show that this algorithm is conditionally optimal, i.e. it cannot be derandomised unless polynomial identity testing is in

. For

-definable graph classes

of bounded pathwidth, we improve the previous complexity upper bound for

from

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

-complete.