Algorithms and Complexity of Range Clustering

TL;DR

This paper introduces range-based clustering criteria inspired by image segmentation, demonstrating that these problems are generally easier and can lead to efficient algorithms compared to traditional similarity-based clustering.

Contribution

The paper proposes new range-based objective functions for clustering, showing their computational advantages and potential as efficient alternatives to NP-hard similarity-based methods.

Findings

Range-based clustering problems are generally easier than similarity-sum problems.

Several proposed range-based objectives can be optimized efficiently.

Range criteria are motivated by practical image segmentation applications.

Abstract

We introduce a novel criterion in clustering that seeks clusters with limited range of values associated with each cluster's elements. In clustering or classification the objective is to partition a set of objects into subsets, called clusters or classes, consisting of similar objects so that different clusters are as dissimilar as possible. We propose a number of objective functions that employ the range of the clusters as part of the objective function. Several of the proposed objectives mimic objectives based on sums of similarities. These objective functions are motivated by image segmentation problems, where the diameter, or range of values associated with objects in each cluster, should be small. It is demonstrated that range-based problems are in general easier, in terms of their complexity, than the analogous similarity-sum problems. Several of the problems we present could therefore be viable alternatives to existing clustering problems which are NP-hard, offering the advantage of efficient algorithms.