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.

翻译：给定图$G$的\textbf{稀疏化}是旨在近似或保留$G$某些性质的一个更稀疏的图（通常是子图）。稀疏化的例子包括但不限于生成树、斯坦纳树、骨架图、仿真图以及距离保持子图。在所有这些问题中，每个顶点具有相同的优先级。然而，现实世界的图通常为不同顶点分配不同的“优先级”或“级别”，其中更高优先级的顶点需要它们之间更高质量的连通性。斯坦纳树问题的多优先级变体已在先前文献中得到研究，但这种推广在其他稀疏化问题中研究得少得多。在本文中，我们定义了一个广义的多优先级问题，并提出了一种可适用于多种图稀疏化的向上取整方法。我们的分析提供了一种系统化方法，在拥有单优先级子程序的情况下，可以为广泛图稀疏化问题的多优先级变体计算近似解。