Graph Sparsification with Priority

Abstract

Given a graph, a \emph{sparsification} is a smaller graph which approximates or preserves some property of the original graph. Well-known sparsifications include but are not limited to spanning trees (preserving connectivity), Steiner trees and forests (preserving connectivity between certain vertices or vertex pairs), spanners and emulators (preserving distances up to given error), distance preservers, cut and spectral sparsifiers, and variants of these. Sparsifications have application in many network design problems including multimedia and electrical power distribution, multicast routing, motion planning, computing approximate all-pairs shortest paths, computational biology, and graph visualization. We first develop several theoretical results involving additive spanners in weighted graphs. An additive $+\beta$ spanner is a subgraph which preserves distances in the original graph up to $+\beta$ additive error; such spanners were largely only studied in unweighted graphs. By generalizing an initialization technique and neighborhood lemma, additive spanners in weighted graphs can be constructed with nearly the same size bounds as in unweighted graphs. We then develop a similar neighborhood lemma to construct \emph{lightweight} additive spanners which aim to minimize total edge weight instead of number of edges. One persistent issue in the above sparsifications involves the underlying graph itself: important vertices (such as traffic hubs in a transportation network, or influential users in a social network) are essentially indistinguishable from less important vertices. We then consider multi-priority generalizations of the above problems where vertices possess different priority, level, or Quality of Service (QoS) requirements, in which higher-priority vertices require higher-quality connections between them. We mainly study this generalization applied to the Steiner tree and node-weighted Steiner tree problems, but this notion can be extended to other sparsifications including spanners. Several approximation algorithms for the priority Steiner tree problem (both edge- and node-weighted) which return a solution within a logarithmic factor are discussed. Open problems include closing the approximability gap or developing more efficient approximation algorithms for the above priority Steiner tree problems, improving the lightness bounds related to weighted additive spanners, and constructing spanners which are simultaneously sparse and lightweight.