On the Approximability of the Single Allocation p-Hub Center Problem with Parameterized Triangle Inequality

For some $$\beta \ge 1/2$$ , a $$\varDelta _{\beta }$$ -metric graph $$G=(V,E,w)$$ is complete edge-weighted 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 vertices $$u,v,x\in V$$ . A $$H=(V', E')$$ called spanning subgraph of $$G=(V, E)$$ if $$V'=V$$ $$E'\subseteq E$$ Given positive integer p, let H be G satisfying the three conditions: (i) there exists vertex subset $$C\subseteq C forms clique size p in H; (ii) set $$V \setminus C$$ an independent (iii) each $$v\in V adjacent to exactly one C. The are hubs $$V\setminus non-hubs. }\text {-}p$$ -Hub Center Problem ( HCP) find conditions diameter minimized. In this paper, we study } \text {-} p$$ HCP \frac{1}{2}$$ We show any $$\epsilon >0$$ approximate ratio $$g(\beta )-\epsilon $$ NP-hard give $$r(\beta )$$ -approximation algorithms same problem where functions $$\frac{3-\sqrt{3}}{2}<\beta \frac{5+\sqrt{5}}{10}$$ approximation algorithm reaches lower bound )= \frac{3\beta -2\beta ^2}{3(1-\beta )}$$ $$\frac{3-\sqrt{3}}{2} < \frac{2}{3}$$ ) = +\beta ^2$$ $$\frac{2}{3}\le $$\frac{5+\sqrt{5}}{10}\le 1$$ =\frac{4\beta ^2+3\beta -1}{5\beta -1}$$ \min \{\beta ^2, \frac{4\beta ^2+5\beta +1}{5\beta +1}\}$$ Additionally, -1}{3\beta )=\min \{\frac{\beta ^2+4\beta }{3},2\beta \}$$ 2$$ upper on =2\beta linear \frac{3 - \sqrt{3}}{2}$$ )=r(\beta )=1$$ i.e., polynomial-time solvable.