The Approximability of the 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 an integer p, the \(\varDelta _{\beta }\)-p Hub Center Problem (\(\varDelta _{\beta }\)-pHCP) is to find a spanning subgraph \(H^{*}\) of G such that (i) vertices (hubs) in \(C^{*}{\subset }V\) form a clique of size p in \(H^{*}\); (ii) vertices (non-hubs) in \(V{\setminus }C^{*}\) form an independent set in \(H^{*}\); (iii) each non-hub \(v\in V{\setminus }C^{*}\) is adjacent to exactly one hub in \(C^{*}\); and (iv) the diameter \(D(H^{*})\) is minimized. For \(\beta = 1\), \(\varDelta _{\beta }\)-pHCP is NP-hard. (Chen et al., CMCT 2016) proved that for any \(\varepsilon >0\), it is NP-hard to approximate the \(\varDelta _{\beta }\)-pHCP to within a ratio \(\frac{4}{3}-\varepsilon \) for \(\beta = 1\). In the same paper, a \(\frac{5}{3}\)-approximation algorithm was given for \(\varDelta _{\beta }\)-pHCP for \(\beta = 1\). In this paper, we study \(\varDelta _{\beta }\)-pHCP for all \(\beta \ge \frac{1}{2}\). We show that for any \(\varepsilon > 0\), to approximate the \(\varDelta _{\beta }\)-pHCP 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., \(\varDelta _{\beta }\)-pHCP is polynomial time solvable. If \(\frac{3-\sqrt{3}}{2} < \beta \le \frac{2}{3}\), we have \(r(\beta ) = g(\beta ) = \frac{3\beta - 2\beta ^2}{3(1-\beta )}\). For \(\frac{2}{3} \le \beta \le \frac{5+\sqrt{5}}{10}\), \(r(\beta )=g(\beta ) = \beta +\beta ^2\). Moreover, for \(\beta \ge 1\), we have \(g(\beta ) = \beta \cdot \frac{4\beta - 1}{3\beta -1}\) and \(r(\beta ) = 2\beta \), the approximability of the problem (i.e., upper and lower bound) is linear in \(\beta \).