Clustering a 2d pareto front: P-center problems are solvable in polynomial time

Abstract

Having many non dominated solutions in bi-objective optimization problems, this paper aims to cluster the Pareto front using Euclidean distances. The p-center problems, both in the discrete and continuous versions, become solvable with a dynamic programming algorithm. Having N points, the complexity of clustering is O(K N log N) (resp. O(K N log2 N)) time and O(N) memory space for the continuous (resp. discrete) K-center problem for K ≥ 3, and in O(N log N) time for such 2-center problems. Furthermore, parallel implementations allow quasi-linear speed-up for the practical applications.