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.
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Dupin, N., Nielsen, F., Talbi, EG. (2020). K-Medoids Clustering Is Solvable in Polynomial Time for a 2d Pareto Front. In: Le Thi, H., Le, H., Pham Dinh, T. (eds) Optimization of Complex Systems: Theory, Models, Algorithms and Applications. WCGO 2019. Advances in Intelligent Systems and Computing, vol 991. Springer, Cham. https://doi.org/10.1007/978-3-030-21803-4_79
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