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

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 \(\beta \).