K-Medoids Clustering Is Solvable in Polynomial Time for a 2d Pareto Front

Abstract

The k-medoids problem is a discrete sum-of-square clustering problem, which is known to be more robust to outliers than k-means clustering. As an optimization problem, k-medoids is NP-hard. This paper examines k-medoids clustering in the case of a two-dimensional Pareto front, as generated by bi-objective optimization approaches. A characterization of optimal clusters is provided in this case. This allows to solve k-medoids to optimality in polynomial time using a dynamic programming algorithm. More precisely, having N points to cluster, the complexity of the algorithm is proven in $$O(N^3)$$ time and $$O(N^2)$$ memory space. This algorithm can also be used to minimize conjointly the number of clusters and the dissimilarity of clusters. This bi-objective extension is also solvable to optimality in $$O(N^3)$$ time and $$O(N^2)$$ memory space, which is useful to choose the appropriate number of clusters for the real-life applications. Parallelization issues are also discussed, to speed-up the algorithm in practice.