Formulation of 3-clustering as a 3-SAT problem

Abstract

Cluster analysis is a sub-field in artificial intelligence and machine learning that refers to grouping of objects based on some objective metric. Clustering with constraints is an active area of research in artificial intelligence. Adding constraints to clustering improves the performance of a variety of algorithms. Cluster analysis is concerned with the problem of partitioning a given set of entities into homogeneous and well-separated subsets called clusters. Cluster Analysis aims at finding subsets, called clusters, which are homogeneous and/or well separated. Minimum sum of diameters clustering for two Cluster can be solved by reduction constraints into the 2-Conjunctive Normal Form statement. Hansen provided an algorithm that solved minimum sum of diameter problem for two clusters that run with time complexity O (n 3logn). For three or more clusters the problem of determining a minimum diameter partition is NP-complete. However 2-cluster problem is solvable in polynomial time. This paper explains what are the constraints in three cluster analysis and this paper also gives a formulation in-fact a technique to represent the constraints of 3-cluster in 3-Conjunctive Normal Form statement.