Given an undirected graph $G=(V,E)$ with a nonnegative edge length function and an integer $p$, $0 < p < |V|$, the $p$-centdian problem is to find $p$ vertices (called the {\it centdian set}) of $V$ such that the {\it eccentricity} plus {\it median-distance} is minimized, in which the {\it eccentricity} is the maximum (length) distance of all vertices to their nearest {\it centdian set} and the {\it median-distance} is the total (length) distance of all vertices to their nearest {\it centdian set}. The {\it eccentricity} plus {\it median-distance} is called the {\it centdian-distance}. The purpose of the $p$-centdian problem is to find $p$ open facilities (servers) which satisfy the quality-of-service of the minimum total distance ({\it median-distance}) and the maximum distance ({\it eccentricity}) to their service customers, simultaneously. If we converse the two criteria, that is given the bound of the {\it centdian-distance} and the objective function is to minimize the cardinality of the {\it centdian set}, this problem is called the converse centdian problem. In this paper, we prove the $p$-centdian problem is NP-Complete. Then we design the first non-trivial brute force exact algorithms for the $p$-centdian problem and the converse centdian problem, respectively. Finally, we design two approximation algorithms for both problems.

翻译：以非方向图形 $G= (V,E) 美元,具有非负向边缘功能和整数美元, $0 < p < {V} 美元, 美元中程} 问题在于找到$P$的顶点(长度) (称为 $it cardian set} ), 美元中途, 美元中途 。 美元问题的目的是找到$p( 储量), 满足最低总距离( 美元中程) 的最大( 长度) ; 美元中途, 美元中程 问题在于所有顶点的总( 美元中程), 美元中程 问题在于所有顶点的总( 长度) 距离( 长度) ; 所有顶点的顶点( 美元中位 ) 的顶点( 美元中程值 ) 。 美元问题在于 最低总距离( 美元中程 ) 的顶点( ) 和 最远点( 我们的中间值 ), 的中间值 问题在于 。