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.