We initiate the study of coresets for clustering in graph metrics, i.e., the
shortest-path metric of edge-weighted graphs. Such clustering problems are
essential to data analysis and used for example in road networks and data
visualization. A coreset is a compact summary of the data that approximately
preserves the clustering objective for every possible center set, and it offers
significant efficiency improvements in terms of running time, storage, and
communication, including in streaming and distributed settings. Our main result
is a near-linear time construction of a coreset for k-Median in a general graph
G, with size Oϵ,k​(tw(G)) where tw(G) is the
treewidth of G, and we complement the construction with a nearly-tight size
lower bound. The construction is based on the framework of Feldman and Langberg
[STOC 2011], and our main technical contribution, as required by this
framework, is a uniform bound of O(tw(G)) on the shattering
dimension under any point weights. We validate our coreset on real-world road
networks, and our scalable algorithm constructs tiny coresets with high
accuracy, which translates to a massive speedup of existing approximation
algorithms such as local search for graph k-Median