Let \(G = (V, E, w)\) be an undirected connected edge-weighted graph, where V is the n-vertices set, E is the m-edges set, and \(w: E \rightarrow \mathbb {R}^+\) is a positive edge-weight function. Given \(G = (V, E, w)\), a subset \(X \subseteq V\) of p terminals, and a subset \(F \subseteq E\) of candidate edges, the Absolute 1-Center Problem (A1CP) aims to find a point on some edge in F to minimize the distance from it to X. This paper revisits this classic and polynomial-time solvable problem from a novel perspective and finds some new and nontrivial properties of it, with the highlight of establishing the relationship between the A1CP and the saddle point of distance matrix. In this paper, we prove that an absolute 1-center is just a vertex 1-center if the all-pairs shortest paths distance matrix from the vertices covered by the candidate edges in F to X has a (global) saddle point. Furthermore, we define the local saddle point of edge and demonstrate that we can sift the candidate edge having a local saddle point. By incorporating the method of sifting edges into the framework of the well-known Kariv and Hakimi’s algorithm, we develop an \(O(m + p m^*+ n p \log p)\)-time algorithm for A1CP, where \(m^*\) is the number of the remaining candidate edges. Specifically, it takes \(O(m^*n + n^2 \log n)\) time to apply our algorithm to the classic A1CP when the distance matrix is known and \(O(m n + n^2 \log n)\) time when the distance matrix is unknown, which are smaller than \(O(mn + n^2 \log n)\) and \(O(mn + n^3)\) of Kariv and Hakimi’s algorithm, respectively.

令\(G = (V, E, w)\)是一个无向连通边加权图，其中V是n 个顶点集合，E是m个边集合，并且\(w: E \rightarrow \mathbb {R}^+\)是一个正边权函数。给定\(G = (V, E, w)\)，p个终端的子集\(X \subseteq V\)和候选边的子集\(F \subseteq E\) ，绝对 1 中心问题(A1CP) 旨在在F的某个边上找到一个点，以最小化它到X的距离. 本文从新颖的角度重新审视了这个经典的多项式时间可解问题，并发现了它的一些新的和非平凡的性质，重点是建立 A1CP 与距离矩阵鞍点之间的关系。在本文中，我们证明如果从F到X中候选边覆盖的顶点的所有对最短路径距离矩阵具有（全局）鞍点，则绝对 1 中心只是顶点 1 中心。此外，我们定义局部边缘的鞍点并证明我们可以筛选具有局部鞍点的候选边缘。通过将筛选边缘的方法结合到著名的 Kariv 和 Hakimi 算法的框架中，我们为 A1CP 开发了一个\(O(m + pm^*+ np \log p)\)时间算法，其中\(m ^*\)是剩余候选边的数量。具体来说，当距离矩阵已知且\(O(mn + n^2 \log n )时，将我们的算法应用于经典 A1CP需要\(O(m^*n + n^2 \log n)\)时间)\)距离矩阵未知时的时间，分别小于Kariv 和 Hakimi 算法的\(O(mn + n^2 \log n)\)和\(O(mn + n^3)\) .