In this paper, we consider the (weighted) one-center problem of uncertain points on a cactus graph. Given are a cactus graph $G$ and a set of $n$ uncertain points. Each uncertain point has $m$ possible locations on $G$ with probabilities and a non-negative weight. The (weighted) one-center problem aims to compute a point (the center) $x^*$ on $G$ to minimize the maximum (weighted) expected distance from $x^*$ to all uncertain points. No previous algorithm is known for this problem. In this paper, we propose an $O(|G| + mn\log mn)$-time algorithm for solving it. Since the input is $O(|G|+mn)$, our algorithm is almost optimal.

翻译：本文研究了仙人掌图上不确定点的（加权）单中心问题。给定一个仙人掌图$G$以及一组$n$个不确定点。每个不确定点在$G$上具有$m$个可能的位置，每个位置对应一个概率和一个非负权重。（加权）单中心问题的目标是计算$G$上的一个点（中心）$x^*$，以最小化从$x^*$到所有不确定点的最大（加权）期望距离。此前尚无针对该问题的已知算法。本文提出了一种时间复杂度为$O(|G| + mn\log mn)$的求解算法。由于输入规模为$O(|G|+mn)$，该算法几乎是时间最优的。