We consider the following problem about dispersing points. Given a set of points in the plane, the task is to identify whether by moving a small number of points by small distance, we can obtain an arrangement of points such that no pair of points is ``close" to each other. More precisely, for a family of $n$ points, an integer $k$, and a real number $d > 0$, we ask whether at most $k$ points could be relocated, each point at distance at most $d$ from its original location, such that the distance between each pair of points is at least a fixed constant, say $1$. A number of approximation algorithms for variants of this problem, under different names like distant representatives, disk dispersing, or point spreading, are known in the literature. However, to the best of our knowledge, the parameterized complexity of this problem remains widely unexplored. We make the first step in this direction by providing a kernelization algorithm that, in polynomial time, produces an equivalent instance with $O(d^2k^3)$ points. As a byproduct of this result, we also design a non-trivial fixed-parameter tractable (FPT) algorithm for the problem, parameterized by $k$ and $d$. Finally, we complement the result about polynomial kernelization by showing a lower bound that rules out the existence of a kernel whose size is polynomial in $k$ alone, unless $\mathsf{NP} \subseteq \mathsf{coNP}/\text{poly}$.

翻译：考虑如下关于分散点集的问题：给定平面上的点集，任务是判断能否通过少量移动少量点（移动距离不超过给定值）来获得一个所有点对均不"邻近"的布局。具体而言，对于包含$n$个点的点集、整数$k$和实数$d > 0$，我们询问是否存在一种方案，使得最多移动$k$个点（每个点移动距离不超过$d$），且最终任意两点间距离至少为某个固定常数（例如$1$）。文献中已知多种针对该问题变体的近似算法，其命名包括"远距离代表点"、"圆盘分散"或"点扩散"等。然而据我们所知，该问题的参数复杂度仍鲜有探索。我们通过设计一个多项式时间内生成等价实例（包含$O(d^2k^3)$个点）的核化算法，在这一方向上迈出第一步。作为该结果的副产品，我们还为问题设计了以$k$和$d$为参数的非平凡固定参数可解（FPT）算法。最后，我们通过下界证明补充多项式核化结果：除非$\mathsf{NP} \subseteq \mathsf{coNP}/\text{poly}$，否则不存在核大小仅为$k$的多项式函数规模的核。