On Undecided LP, Clustering and Active Learning

TL;DR

This paper explores algorithms for clustering and geometric covering problems with limited color and point information, providing efficient query-based methods and the first near-linear time approximation for terrain simplification.

Contribution

It introduces query-efficient algorithms for monochromatic clustering and geometric covering, including the first near-linear time approximation for terrain simplification.

Findings

Constant cluster count allows monochromatic clustering with polylogarithmic queries.

Developed the first near-linear time approximation algorithm for terrain simplification.

Achieved subquadratic complexity for terrain algorithms with small complexity.

Abstract

We study colored coverage and clustering problems. Here, we are given a colored point set where the points are covered by (unknown) $k$ clusters, which are monochromatic (i.e., all the points covered by the same cluster, have the same color). The access to the colors of the points (or even the points themselves) is provided indirectly via various queries (such as nearest neighbor, or separation queries). We show that if the number of clusters is a constant, then one can correctly deduce the color of all the points (i.e., compute a monochromatic clustering of the points) using a polylogarithmic number of queries. We investigate several variants of this problem, including Undecided Linear Programming, covering of points by $k$ monochromatic balls, covering by $k$ triangles/simplices, and terrain simplification. For the later problem, we present the first near linear time approximation algorithm. While our approximation is slightly worse than previous work, this is the first algorithm to have subquadratic complexity if the terrain has "small" complexity.