  In this paper, we investigate the relative dominant dimension with respect to
an injective module and characterize the algebras with finite relative dominant
dimension. As an application, we introduce the almost n-precluster tilting
module and establish a correspondence between almost n-precluster tilting
modules and almost n-minimal Auslander-Gorenstein algebras. Moreover, we give a
description of the Gorenstein projective modules over almost n-minimal
Auslander-Gorenstein algebras in terms of the corresponding almost n-precluster
tilting modules.
