  A \emph{sparsification} of a given graph $G$ is a sparser graph (typically a
subgraph) which aims to approximate or preserve some property of $G$. Examples
of sparsifications include but are not limited to spanning trees, Steiner
trees, spanners, emulators, and distance preservers. Each vertex has the same
priority in all of these problems. However, real-world graphs typically assign
different ``priorities'' or ``levels'' to different vertices, in which
higher-priority vertices require higher-quality connectivity between them.
Multi-priority variants of the Steiner tree problem have been studied in prior
literature but this generalization is much less studied for other
sparsification problems. In this paper, we define a generalized multi-priority
problem and present a rounding-up approach that can be used for a variety of
graph sparsifications. Our analysis provides a systematic way to compute
approximate solutions to multi-priority variants of a wide range of graph
sparsification problems given access to a single-priority subroutine.
