Multi-Priority Graph Sparsification

TL;DR

This paper introduces a generalized multi-priority framework for graph sparsification problems, allowing for approximate solutions that respect different priority levels of vertices, extending beyond traditional single-priority approaches.

Contribution

It defines a new multi-priority problem framework and offers a rounding-up method applicable to various graph sparsification problems, enhancing flexibility and applicability.

Findings

Provides a systematic approach for multi-priority graph sparsification

Enables approximation solutions using a single-priority subroutine

Extends sparsification techniques to prioritize vertices by importance

Abstract

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.