IWOCA 2018: Combinatorial Algorithms pp 115-127

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

• Li-Hsuan Chen
• Sun-Yuan Hsieh
• Ling-Ju Hung
• Ralf Klasing
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10979)

## 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.

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© Springer International Publishing AG, part of Springer Nature 2018

## Authors and Affiliations

• Li-Hsuan Chen
• 1
• Sun-Yuan Hsieh
• 2
• Ling-Ju Hung
• 1
Email author
• Ralf Klasing
• 3
1. 1.AROBOT Innovation CO., LTD.New Taipei CityTaiwan
2. 2.Department of Computer Science and Information EngineeringNational Cheng Kung UniversityTainanTaiwan
3. 3.CNRS, LaBRI, Université de BordeauxTalence cedexFrance