k-medoids and p-median clustering are solvable in polynomial time for a 2d Pareto front

Nicolas Dupin; Frank Nielsen; El-Ghazali Talbi

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arXiv · 1806.02098

k-medoids and p-median clustering are solvable in polynomial time for a 2d Pareto front

Nicolas Dupin·Frank Nielsen·El-Ghazali Talbi

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Abstract

This paper examines a common extension of k-medoids and k-median clustering in the case of a two-dimensional Pareto front, as generated by bi-objective optimization approaches. A characterization of optimal clusters is provided, which allows to solve the optimization problems to optimality in polynomial time using a common dynamic programming algorithm. More precisely, having $N$ points to cluster in $K$ subsets, the complexity of the algorithm is proven in $O(N^3)$ time and $O(K.N)$ memory space when $K\geqslant 3$, cases $K=2$ having a time complexity in $O(N^2)$. Furthermore, speeding-up the dynamic programming algorithm is possible avoiding useless computations, for a practical speed-up without improving the complexity. Parallelization issues are also discussed, to speed-up the algorithm in practice.

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BibTeX RIS

Nicolas Dupin, Frank Nielsen, El-Ghazali Talbi. 2018-06-06. k-medoids and p-median clustering are solvable in polynomial time for a 2d Pareto front. https://arxiv.org/abs/1806.02098

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