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

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 $G=(V, E, w)$, 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.

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. The main work for this article was done while Li-Hsuan Chen and Ling-Ju Hung (corresponding author) were with the National Cheng Kung University.