Approaches to minimum sum of diameter and radii clustering

Abstract

Cluster analysis is a key technique in the data analysis and is being applied in a variety of engineering and scientific disciplines such as biology, medicine, marketing, information retrieval and pattern recognition. Stereotyped clustering method propagates dissection effect. To thwart dissection effect, many real-time applications uses minimum sum of diameter clustering (MSDC) or minimum sum of radii clustering (MSRC). As, MSDC and MSRC problems are NP-complete so it is natural to seek approximation algorithms with the best provable approximation ratio. This thesis presents approaches for generating solutions for clustering problems. The thesis addresses clustering problems in three parts: (i) Comparative review of clustering algorithms (ii) SAT formulation-based approach, and (iii) Constraint based clustering approach In the first part of this thesis: we investigated approach, distance criterion, optimization methods, geometrical properties, assumptions, issues, limitations, and time-bound to propose a comprehensive and comparative analysis of algorithmic complexity of various exact and approximation clustering algorithms for Euclidean, metric and geometric version of MSDC and MSRC problem. SAT formulation-based approach: In this approach, we investigate the technique for the reduction of the 3-cluster problem into 3-SAT and kcluster problem into k-SAT. In the constraint based clustering approach, we address three types of problems: 3-clustering problem for minimum sum of diameter, word clustering algorithm based on the k-clustering algorithm, and constraint based approximation algorithm for the min-cost k-cover problem. In the 3-clustering problem, we proposed a constraint algorithm for three clustering for the minimum sum of diameter problem. In the word clustering algorithm, a new constraint word clustering algorithm is proposed. The main challenge is to find out constraint for words having an asymmetric relationship between them. In this context, we investigated properties of word cloud like symmetric, transitive, and implicative and also investigated various types of associations like strong, weak, and zero association between words. We proposed a constraint word algorithm based on the investigated properties and association. Furthermore, we bring the concept of constraints in the min-cost k-cover problem to improve the performance. Constraints help in reducing the number of distinct maximal discs. By incorporating the constraints into the min-cost kcover algorithm, we proposed an approximation algorithm with the improved approximation ratio.