The $p$-center problem is finding the location of $p$ facilities among a set of $n$ demand points such that the maximum distance between any demand point and its nearest facility is minimized. In this paper, we study this problem in the context of uncertainty, that is, the location of the demand points may change in a region like a disk or a segment, or belong to a finite set of points. We introduce Max-$p$-center and Min-$p$-center problems which are the worst and the best possible solutions for the $p$-center problem under such locational uncertainty. We propose approximation and parameterized algorithms to solve these problems under the Euclidean metric. Further, we study the MinMax Regret 1-center problem under uncertainty and propose a linear-time algorithm to solve it under the Manhattan metric as well as an $O\left(n^4\right)$ time algorithm under the Euclidean metric.

这$p$-中心问题是找到的位置$p$一套间的设施$n$请求点，使得任何请求点与其最近设施点之间的最大距离最小化。在本文中，我们在不确定性的背景下研究这个问题，即需求点的位置可能在一个区域（如磁盘或段）中发生变化，或者属于有限的点集。我们介绍Max-$p$-中心和最小-$p$-最坏和最好的可能解决方案的中心问题$p$- 这种位置不确定性下的中心问题。我们提出了近似和参数化算法来解决欧几里得度量下的这些问题。此外，我们研究了不确定性下的 MinMax Regret 1-center 问题，并提出了一种线性时间算法来解决它在曼哈顿度量下的问题以及一个$欧\left(n^{\mathrm{4个}}\right)$欧几里德度量下的时间算法。