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

  • Nicolas DupinEmail author
  • Frank Nielsen
  • El-Ghazali Talbi
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)


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.


Bi-objective optimization Clustering algorithms K-medoids Euclidean sum-of-squares clustering Pareto front Dynamic programming Bi-objective clustering 


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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Nicolas Dupin
    • 1
    Email author
  • Frank Nielsen
    • 2
  • El-Ghazali Talbi
    • 3
  1. 1.LRI, Université Paris-Sud, Université Paris-SaclayParisFrance
  2. 2.Sony Computer Science Laboratories Inc.TokyoJapan
  3. 3.Univ. Lille, UMR 9189 - CRIStAL - Centre de Recherche en Informatique Signal et Automatique de LilleLilleFrance

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