On the Complexity of the Star p-hub Center Problem with Parameterized Triangle Inequality

Abstract

A complete weighted graph $G= (V, E, w)$ is called $\varDelta _{\beta }$-metric, for some $\beta \ge 1/2$, if G satisfies the $\beta$-triangle inequality, i.e., $w(u,v) \le \beta \cdot (w(u,x) + w(x,v))$ for all vertices $u,v,x\in V$. Given a $\varDelta _{\beta }$-metric graph $G=(V, E, w)$ and a center $c\in V$, and an integer p, the $\varDelta _{\beta }$-Star p-Hub Center Problem ($\varDelta _{\beta }$-SpHCP) is to find a depth-2 spanning tree T of G rooted at c such that c has exactly p children and the diameter of T is minimized. The children of c in T are called hubs. For $\beta = 1$, $\varDelta _{\beta }$-SpHCP is NP-hard. (Chen et al., COCOON 2016) proved that for any $\varepsilon >0$, it is NP-hard to approximate the $\varDelta _{\beta }$-SpHCP to within a ratio $1.5-\varepsilon$ for $\beta = 1$. In the same paper, a $\frac{5}{3}$-approximation algorithm was given for $\varDelta _{\beta }$-SpHCP for $\beta = 1$. In this paper, we study $\varDelta _{\beta }$-SpHCP for all $\beta \ge \frac{1}{2}$. We show that for any $\varepsilon > 0$, to approximate the $\varDelta _{\beta }$-SpHCP to a ratio $g(\beta ) - \varepsilon$ is NP-hard and we give $r(\beta )$-approximation algorithms for the same problem where $g(\beta )$ and $r(\beta )$ are functions of $\beta$. If $\beta \le \frac{3 - \sqrt{3}}{2}$, we have $r(\beta ) = g(\beta ) = 1$, i.e., $\varDelta _{\beta }$-SpHCP is polynomial time solvable. If $\frac{3-\sqrt{3}}{2} < \beta \le \frac{2}{3}$, we have $r(\beta ) = g(\beta ) = \frac{1 + 2\beta - 2\beta ^2}{4(1-\beta )}$. For $\frac{2}{3} \le \beta \le 1$, $r(\beta ) = \min \{\frac{1 + 2\beta - 2\beta ^2}{4(1-\beta )}, 1 + \frac{4\beta ^2}{5\beta +1}\}$. Moreover, for $\beta \ge 1$, we have $r(\beta ) = \min \{\beta + \frac{4\beta ^2- 2\beta }{2 + \beta }, 2\beta + 1\}$. For $\beta \ge 2$, the approximability of the problem (i.e., upper and lower bound) is linear in $\beta$.

Parts of this research were supported by the Ministry of Science and Technology of Taiwan under grants MOST 105–2221–E–006–164–MY3, MOST 103–2218–E–006–019–MY3, and MOST 103-2221-E-006-135-MY3. L.-H. Chen, L.-J. Hung (corresponding author), and C.-W. Lee are supported by the Ministry of Science and Technology of Taiwan under grants MOST 105–2811–E–006–071, –046, and –022, respectively.

R. Klasing–Part of this work was done while Ralf Klasing was visiting the Department of Computer Science and Information Engineering at National Cheng Kung University. This study has been carried out in the frame of the “Investments for the future” Programme IdEx Bordeaux - CPU (ANR-10-IDEX-03-02). Research supported by the LaBRI under the “Projets émergents” program.