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

Abstract

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