Coresets for constrained k-median and k-means clustering in low dimensional Euclidean space
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We study (Euclidean) k-median and k-means with constraints in the streaming model. There have been recent efforts to design unified algorithms to solve constrained k-means problems without using knowledge of the specific constraint at hand aside from mild assumptions like the polynomial computability of feasibility under the constraint (compute if a clustering satisfies the constraint) or the presence of an efficient assignment oracle (given a set of centers, produce an optimal assignment of points to the centers which satisfies the constraint). These algorithms have a running time exponential in k, but can be applied to a wide range of constraints. We demonstrate that a technique proposed in 2019 for solving a specific constrained streaming k-means problem, namely fair k-means clustering, actually implies streaming algorithms for all these constraints. These work for low dimensional Euclidean space. [Note that there are more algorithms for streaming fair k-means today, in particular they exist for high dimensional spaces now as well.]
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