Planar p-center problems are solvable in polynomial time when clustering a Pareto Front

TL;DR

This paper demonstrates that clustering Pareto fronts in bi-objective optimization using p-center problems can be solved efficiently in polynomial time with a dynamic programming approach, enabling faster multi-objective heuristics.

Contribution

It introduces a polynomial-time dynamic programming algorithm for solving discrete and continuous p-center clustering problems on Pareto fronts, with proven complexity bounds.

Findings

Polynomial-time algorithms for clustering Pareto fronts.

Complexity of O(KN log N) for continuous and O(KN log^2 N) for discrete p-center problems.

Applicability to multi-objective heuristics for Pareto front approximation.

Abstract

This paper is motivated by real-life applications of bi-objective optimization. Having many non dominated solutions, one wishes to cluster the Pareto front using Euclidian distances. The p-center problems, both in the discrete and continuous versions, are proven solvable in polynomial time with a common dynamic programming algorithm. Having $N$ points to partition in $K\geqslant 3$ clusters, the complexity is proven in $O(KN\log N)$ (resp $O(KN\log^2 N)$) time and $O(KN)$ memory space for the continuous (resp discrete) $K$-center problem. $2$-center problems have complexities in $O(N\log N)$. To speed-up the algorithm, parallelization issues are discussed. A posteriori, these results allow an application inside multi-objective heuristics to archive partial Pareto Fronts.