Separated high-density points and valley detection on polynomial densities are ∃R-complete; connected components and hole detection are ∃R-hard, giving the first exact classification of continuous clustering in the real polynomial hierarchy.
The hardness ofk-means clustering.Technical Report CS2008-0916, UC San Diego
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How Hard Is Continuous Clustering? Lower Bounds from the Existential Theory of the Reals
Separated high-density points and valley detection on polynomial densities are ∃R-complete; connected components and hole detection are ∃R-hard, giving the first exact classification of continuous clustering in the real polynomial hierarchy.