Approximation Algorithms for the p-Hub Center Routing Problem in Parameterized Metric Graphs

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 \(G=(V, E, w)\), the Single Allocation at most p-Hub Center Routing problem is to find a spanning subgraph \(H^{*}\) of G such that (i) any pair of vertices in \(C^{*}\) is adjacent in \(H^{*}\) where \(C^{*}\subset V\) and \(|C^{*}|\le p\); (ii) any pair of vertices in \(V\setminus C^{*}\) is not adjacent in \(H^{*}\); (iii) each \(v\in V\setminus C^{*}\) is adjacent to exactly one vertex in \(C^{*}\); and (iv) the routing cost \(r(H^{*}) = \sum _{u,v\in V} d_{H^{*}}(u,v)\) is minimized where \(d_{H^{*}}(u,v)= w(u,f^{*}(u))+ w(f^{*}(u),f^{*}(v))+ w(v,f^{*}(v))\) and \(f^{*}(u),f^{*}(v)\) are the vertices in \(C^{*}\) adjacent to u and v in \(H^{*}\), respectively. Note that \(w(v,f^{*}(v)) = 0\) if \(v\in C^{*}\). The vertices selected in \(C^{*}\) are called hubs and the rest of vertices are called non-hubs. In this paper, we show that the Single Allocation at most p-Hub Center Routing problem is NP-hard in \(\varDelta _{\beta }\)-metric graphs for any \(\beta > 1/2\). Moreover, we give \(2\beta \)-approximation algorithms running in time \(O(n^2)\) for any \(\beta > 1/2\) where n is the number of vertices in the input graph.