Let $G=(V,E,w)$ be a ${\mathrm{\Delta }}_{\beta }$-metric graph with a distance function $w(\cdot ,\cdot )$ on V such that $w(v,v)=0$, $w(u,v)=w(v,u)$, and $w(u,v)\le \beta \cdot (w(u,x)+w(x,v))$ for all $u,v,x\in V$. Given a positive integer p, let H be a spanning subgraph of G satisfying the conditions that vertices (hubs) in $C\subset V$ form a clique of size at most p in H, vertices (non-hubs) in $V\setminus C$ form an independent set in H, and each non-hub $v\in V\setminus C$ is adjacent to exactly one hub in C. Define $d_H(u,v)=w(u,f(u))+w(f(u),f(v))+w(v,f(v))$ where $f(u)$ and $f(v)$ are hubs adjacent to u and v in H respectively. Notice that if u is a hub in H then $w(u,f(u))=0$. Let $r(H)={\sum }_{u,v\in V}{d}_H(u,v)$ be the routing cost of H. The Single Allocation at most p-Hub Center Routing problem is to find a spanning subgraph H of G such that $r(H)$ is minimized. In this paper, we show that the Single Allocation at most p-Hub Center Routing problem is NP-hard in ${\mathrm{\Delta }}_{\beta }$-metric graphs for any $\beta >1/2$. Moreover, we give 2β-approximation algorithms running in time $O(n^2)$ for any $\beta >1/2$ where n is the number of vertices in the input graph. Finally, we show that the approximation ratio of our algorithms is at least $\mathrm{\Omega }(\beta )$, and we examine the structure of any potential $o(\beta )$-approximation algorithm.

让 $G=（V，Ë，w）$ 成为 ${\mathrm{\Delta }}_{\beta }$距离函数的度量图 $w（\cdot ，\cdot ）$在V上$w（v，v）=0$， $w（ü，v）=w（v，ü）$和 $w（ü，v）\le \beta \cdot （w（ü，X）+w（X，v））$ 对全部 $ü，v，X\in V$。给定一个正整数p，让ħ是一个生成子图ģ满足条件顶点（中心）的$C\subset V$在H处最多形成一个p大小的族群，在H处形成顶点（非中心）$V\setminus C$在H中形成一个独立的集合，每个非中心$v\in V\setminus C$与C中恰好一个集线器相邻。定义$d_H（ü，v）=w（ü，F（ü））+w（F（ü），F（v））+w（v，F（v））$ 哪里 $F（ü）$ 和 $F（v）$邻近轮毂ü和v在ħ分别。请注意，如果u是H的集线器，则$w（ü，F（ü））=0$。让$\mathrm{[R}（H）={\sum }_{ü，v\in V}{d}_H（ü，v）$是路由成本的ħ。在最多一个分配 p -Hub中心路由问题是寻找一个支撑子图^ h的摹这样$\mathrm{[R}（H）$被最小化。在本文中，我们证明最多 p-集线器中心路由问题的单分配是NP-hard${\mathrm{\Delta }}_{\beta }$度量图 $\beta >\mathrm{1个}/2$。此外，我们给出了2个及时运行的β近似算法$Ø（{ñ}^2）$ 对于任何 $\beta >\mathrm{1个}/2$其中n是输入图形中的顶点数。最后，我们证明了我们算法的逼近率至少为$\mathrm{\Omega }（\beta ）$，我们会检查任何潜在的结构 $Ø（\beta ）$-近似算法。