Coresets for Clustering in Geometric Intersection Graphs
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Designing coresets–small-space sketches of the data preserving cost of the solutions within (1±ϵ)-approximate factor–is an important research direction in the study of center-based k-clustering problems, such as k-means or k-median. Feldman and Langberg [STOC'11] have shown that for k-clustering of n points in general metrics, it is possible to obtain coresets whose size depends logarithmically in n. Moreover, such a dependency in n is inevitable in general metrics. A significant amount of recent work in the area is devoted to obtaining coresests whose sizes are independent of n (i.e., “small” coresets) for special metrics, like d-dimensional Euclidean spaces, doubling metrics, metrics of graphs of bounded treewidth, or those excluding a fixed minor. In this paper, we provide the first constructions of small coresets for k-clustering in the metrics induced by geometric intersection graphs, such as Euclidean-weighted Unit Disk/Square Graphs. These constructions follow from a general theorem that identifies two canonical properties of a graph metric sufficient for obtaining small coresets. The proof of our theorem builds on the recent work of Cohen-Addad, Saulpic, and Schwiegelshohn [STOC '21], which ensures small-sized coresets conditioned on the existence of an interesting set of centers, called “centroid set”. The main technical contribution of our work is the proof of the existence of such a small-sized centroid set for graphs that satisfy the two canonical geometric properties. The new coreset construction helps to design the first (1+ϵ)-approximation for center-based clustering problems in UDGs and USGs, that is fixed-parameter tractable in k and ϵ (FPT-AS).
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