Inapproximability of Maximum Diameter Clustering for Few Clusters

Core Concepts

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.

Abstract

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.

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Inapproximability of Maximum Diameter Clustering for Few Clusters

by Henry Fleisc... at arxiv.org 04-08-2024

https://arxiv.org/pdf/2312.02097.pdf

Deeper Inquiries

What are some potential applications of the Max-k-Diameter clustering objective, and how do the inapproximability results impact these applications

The Max-k-Diameter clustering objective has various potential applications in real-world scenarios. One application is in network analysis, where it can be used to identify clusters of closely connected nodes in a network. This can help in understanding the structure of the network, identifying communities, and detecting anomalies. In image processing, Max-k-Diameter clustering can be used for image segmentation, where similar pixels are grouped together based on their distances. This can aid in object recognition, image compression, and pattern recognition tasks. In biology, this clustering objective can be applied to analyze genetic data, identifying groups of genes or proteins with similar characteristics. Additionally, in customer segmentation for marketing purposes, Max-k-Diameter clustering can help in grouping customers based on their behavior or preferences.
The inapproximability results presented in the paper have a significant impact on these applications. The hardness of approximating Max-k-Diameter within a factor of 1.5 in the ℓ1-metric implies that finding an optimal clustering with a guaranteed quality is computationally challenging. This means that in practical applications where exact solutions are not feasible, heuristic or approximation algorithms must be used, potentially leading to suboptimal clustering results. Understanding the limitations of approximation for Max-k-Diameter is crucial for managing expectations and designing efficient algorithms for clustering tasks in various domains.

Can the techniques developed in this paper be extended to prove hardness of approximation for other geometric clustering problems beyond Max-k-Diameter

The techniques developed in this paper, particularly the concept of the r-cloud system for reductions from coloring to Max-k-Diameter, can indeed be extended to prove hardness of approximation for other geometric clustering problems beyond Max-k-Diameter. By constructing suitable cloud systems in different metric spaces and for various hypergraph structures, similar hardness results can be established for related clustering objectives. For instance, by adapting the framework to different hypergraph configurations or metric spaces, one can explore the inapproximability of clustering problems like k-center, k-means, or k-median. This approach provides a versatile method for proving hardness results in geometric optimization problems.

Are there any special cases or restricted input models where improved approximation algorithms for Max-k-Diameter can be obtained, despite the strong hardness results shown in this paper

While the inapproximability results in the paper demonstrate the challenges of approximating Max-k-Diameter within a factor of 1.5 in the ℓ1-metric, there may still be special cases or restricted input models where improved approximation algorithms can be obtained. For example, in scenarios where the pointset exhibits specific geometric properties or structures that allow for efficient clustering, it may be possible to devise specialized algorithms that provide better approximations. Additionally, in low-dimensional spaces or for certain types of data distributions where the clustering task is inherently simpler, tailored algorithms could offer improved performance. By identifying these special cases and leveraging their characteristics, it may be feasible to develop approximation algorithms that outperform the general hardness results shown in the paper.