平面 Pareto 前沿中 p-分散问题的多项式算法
Polynomial algorithms for p-dispersion problems in a planar Pareto Front
AI总结：
本文研究了从2D Pareto Front中选取代表点的p-分散问题，提出了新变体并利用动态规划为多种变体设计了多项式时间算法。
摘要：
本文研究了 p-分散问题，旨在从作为双目标优化解的大型 2D Pareto Front (PF) 中选取 $p\geqslant 2$ 个代表点。考虑了四种标准的 p-分散变体，并为 2D PF 的特定情况引入了一种新变体 Max-Sum-Neighbor p-dispersion。首先，证明了 2D PF 中的 2-dispersion 和 3-dispersion 问题可在 $O(n)$ 时间内求解。其次，为三种 p-分散变体设计了动态规划算法，证明了在 2D PF 中的多项式复杂度。Max-min p-dispersion 可在 $O(pn\log n)$ 时间和 $O(n)$ 内存空间内求解。Max-Sum-Neighbor p-dispersion 证明可在 $O(pn^2)$ 时间和 $O(n)$ 空间内求解。Max-Sum-min p-dispersion 可在 $O(pn^3)$ 时间和 $O(pn^2)$ 空间内求解，该复杂度在 1D 中同样成立，首次证明了 Max-Sum-min p-dispersion 在 1D 中是多项式时间的。此外，讨论了这些算法的特性，以实现高效执行并应用于双目标元启发式算法中。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
