How Hard Is Continuous Clustering? Lower Bounds from the Existential Theory of the Reals
This paper studies the computational difficulty of clustering problems that are defined directly on a continuous probability density. Rather than working with finite samples, we assume the density is given as a polynomial and ask whether it contains certain cluster structures. Four natural questions are examined. First, do there exist several points with high density that are far apart from each other. Second, do two high density points have a midpoint with low density, creating a valley between them. Third, does the region where the density is above a threshold have at least a given number of separate connected pieces. Fourth, does that same region contain a hole, meaning a loop that can...
| Authors | Angshul Majumdar |
|---|---|
| Primary category | cs.CC |
| Published | 24 April 2026 |
| Last updated at source | 24 April 2026 |
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