Polynomial algorithms for p-dispersion problems in a planar Pareto Front

Abstract: In this paper, p-dispersion problems are studied to select $p\geqslant 2$ representative points from a large 2D Pareto Front (PF), solution of bi-objective optimization. Four standard p-dispersion variants are considered. A novel variant, Max-Sum-Neighbor p-dispersion, is introduced for the specific case of a 2D PF. Firstly, $2$-dispersion and $3$-dispersion problems are proven solvable in $O(n)$ time in a 2D PF. Secondly, dynamic programming algorithms are designed for three p-dispersion variants, proving polynomial complexities in a 2D PF. Max-min p-dispersion is solvable in $O(pn\log n)$ time and $O(n)$ memory space. Max-Sum-Neighbor p-dispersion is proven solvable in $O(pn^2)$ time and{$O(n)$} space. Max-Sum-min p-dispersion is solvable in $O(pn^3)$ time and $O(pn^2)$ space, this complexity holds also in 1D, proving for the first time that Max-Sum-min p-dispersion is polynomial in 1D. Furthermore, properties of these algorithms are discussed for an efficient implementation {and for a practical application inside bi-objective meta-heuristics.