Conceitos Básicos
Even when the number of clusters is fixed to 3, the Maximum Diameter Clustering problem is NP-hard to approximate within a factor of 1.5 in the ℓ1-metric and 1.304 in the Euclidean metric.
Resumo
The paper studies the approximability of the Maximum Diameter Clustering (Max-k-Diameter) problem, where the goal is to partition a set of points in a metric space into k clusters such that the maximum pairwise distance between points in the same cluster is minimized.
Key highlights:
The Max-k-Diameter problem was actively studied in the 1980s, with 2-approximation algorithms known for general metrics.
When the number of clusters k is fixed, most popular clustering objectives like k-means, k-median, etc. admit polynomial-time approximation schemes (PTAS). However, the authors show that this is not the case for Max-k-Diameter.
The authors introduce a novel framework called "r-cloud systems" to reduce the panchromatic k-coloring problem on hypergraphs to the approximation version of Max-k-Diameter.
Using this framework, the authors prove that for k ≥ 3, approximating Max-k-Diameter in the ℓ1-metric within a factor better than 1.5 is NP-hard, and in the Euclidean metric within a factor better than 1.304 is NP-hard.
These hardness results hold even when the input pointset is restricted to O(log n) dimensions.
The authors also outline barriers to proving improved hardness of approximation results for Max-k-Diameter.