Coresets for Clustering in Graphs of Bounded Treewidth

We initiate the study of coresets for clustering in graph metrics, i.e., the shortest-path metric of edge-weighted graphs. Such clustering problems (on graph metrics) are essential to data analysis and used for example in road networks and data visualization. Specifically, we consider $(k, z)$-Clustering, where given a metric space $(V, d)$, the goal is to minimize, over all $k$-point center sets $C$, the objective $\sum_{x \in V}{d^z(x, C)}$. This problem is a well-known generalization of both k-Median ($z=1$) and k-Means ($z=2$). A coreset is a compact summary of the data that approximately preserves the clustering objective for every possible center set. Coresets offer 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 of size $O_{\epsilon, k, z}(\mathrm{tw}(G))$ for $(k, z)$-Clustering in a graph $G$ whose treewidth is $\mathrm{tw}(G)$. 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 $O(\mathrm{tw}(G))$ on the shattering dimension under any point weights. Previously, the only construction applicable to graph metrics, even for $z=1$, was a generic one with size $O_{\epsilon, k}(\log n)$ where $n=|V|$ [Feldman and Langberg, STOC 2011]. We complement our construction with an $\Omega_{\epsilon, k}(\mathrm{tw}(G))$ size lower bound, which matches our construction's linear dependence on $\mathrm{tw}(G)$. This further provides the first proof that the $O(\log n)$ factor in the generic upper bound is indeed necessary, and also justifies restricting the graph topology.