Characterization of polynomial asymmetric approximate-homomorphism bounds

Prove that for graphs $F$ and $H$, if there exists an $F$-forest $T$ such that $T\to F$ and $H\to T^{\star}$, then $M_{F,H}(\varepsilon)=\operatorname{poly}(1/\varepsilon)$; equivalently, establish that the stated condition characterizes the cases in which the asymmetric approximate-homomorphism parameter fails to have an exponential lower bound.

Background

The paper constructs graphs G⋆G^{\star} that yield exponential lower bounds for MF,H(ε)M_{F,H}(\varepsilon) whenever FF is 2-connected and HH is not homomorphic to T⋆T^{\star} for any FF-forest TT with T→FT\to F.

The conjecture proposes the converse: whenever some suitable FF-forest TT satisfies H→T⋆H\to T^{\star}, the parameter should have a polynomial upper bound. If true, the construction would be universal for exponential lower bounds and there would be no intermediate growth rate between polynomial and exponential.

References

In fact, we conjecture that the condition in \cref{thm:approx hom hardness} characterises the cases where $M_{F,H}(\varepsilon)$ is at least $2{\poly(1/\varepsilon)}$, and in all other cases $M_{F,H}(\varepsilon)$ is polynomial.

— Asymmetric results about graph homomorphisms  (2502.20278 - Gishboliner et al., 27 Feb 2025) in Conjecture 1.7, Section 1, subsection “Asymmetric approximate homomorphisms”

In fact, we conjecture that the condition in \cref{thm:approx hom hardness} characterises the cases where $M_{F,H}(\varepsilon)$ is at least $2{\poly(1/\varepsilon)}$, and in all other cases $M_{F,H}(\varepsilon)$ is polynomial. Let $F,H$ be graphs, and suppose that there exists an $F$-forest $T$ such that $T\rightarrow F$ and $H\rightarrow T{\star}$. Then $M_{F,H}(\varepsilon) = \poly(1/\varepsilon)$.

— Asymmetric results about graph homomorphisms  (2502.20278 - Gishboliner et al., 27 Feb 2025) in Conjecture 1.9, Section 1.3 (labelled conj:approx homo characterization)