Having many best compromise solutions for bi-objective optimization problems, this paper studies p-dispersion problems to select $p\geqslant 2$ representative points in the Pareto Front(PF). Four standard variants of p-dispersion are considered. A novel variant, denoted Max-Sum-Neighbor p-dispersion, is introduced for the specific case of a 2d PF. Firstly, it is proven that $2$-dispersion and $3$-dispersion problems are 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. The Max-Min p-dispersion problem is proven solvable in $O(pn\log n)$ time and $O(n)$ memory space. The Max-Sum-Min p-dispersion problem is proven solvable in $O(pn^3)$ time and $O(pn^2)$ space. The Max-Sum-Neighbor p-dispersion problem is proven solvable in $O(pn^2)$ time and $O(pn)$ space. Complexity results and parallelization issues are discussed in regards to practical implementation.

中文翻译：

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二维帕累托前沿中 p 色散问题的多项式算法

对双目标优化问题有很多最好的折衷方案，本文研究了p-dispersion问题来选择Pareto Front(PF)中的$p\geqslant 2$代表点。考虑了 p 色散的四种标准变体。针对 2d PF 的特定情况引入了一种新的变体，称为 Max-Sum-Neighbor p-dispersion。首先，证明 $2$-dispersion 和 $3$-dispersion 问题在 2d PF 中的 $O(n)$ 时间内是可以解决的。其次，动态规划算法是为三个 p 色散变体设计的，证明了 2d PF 中的多项式复杂性。Max-Min p-dispersion 问题被证明可以在 $O(pn\log n)$ 时间和 $O(n)$ 内存空间中解决。Max-Sum-Min p-dispersion 问题被证明可以在 $O(pn^3)$ 时间和 $O(pn^2)$ 空间中解决。Max-Sum-Neighbor p-dispersion 问题被证明可以在 $O(pn^2)$ 时间和 $O(pn)$ 空间中解决。在实际实施方面讨论了复杂性结果和并行化问题。