### 摘要

Let G=(V,E,w) be a Δ_{β}-metric graph with a distance function w(⋅,⋅) on V such that w(v,v)=0, w(u,v)=w(v,u), and w(u,v)≤β⋅(w(u,x)+w(x,v)) for all u,v,x∈V. Given a positive integer p, let H be a spanning subgraph of G satisfying the conditions that vertices (hubs) in C⊂V form a clique of size at most p in H, vertices (non-hubs) in V∖C form an independent set in H, and each non-hub v∈V∖C is adjacent to exactly one hub in C. Define d_{H}(u,v)=w(u,f(u))+w(f(u),f(v))+w(v,f(v)) where f(u) and f(v) are hubs adjacent to u and v in H respectively. Notice that if u is a hub in H then w(u,f(u))=0. Let r(H)=∑_{u,v∈V}d_{H}(u,v) be the routing cost of H. The SINGLE ALLOCATION AT MOST p-HUB CENTER ROUTING problem is to find a spanning subgraph H of G such that r(H) is minimized. In this paper, we show that the SINGLE ALLOCATION AT MOST p-HUB CENTER ROUTING problem is NP-hard in Δ_{β}-metric graphs for any β>1/2. Moreover, we give 2β-approximation algorithms running in time O(n^{2}) for any β>1/2 where n is the number of vertices in the input graph. Finally, we show that the approximation ratio of our algorithms is at least Ω(β), and we examine the structure of any potential o(β)-approximation algorithm.

原文 | English |
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頁（從 - 到） | 271-280 |

頁數 | 10 |

期刊 | Theoretical Computer Science |

卷 | 806 |

DOIs | |

出版狀態 | Published - 2020 二月 2 |

### 指紋

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### 引用此

*Theoretical Computer Science*,

*806*, 271-280. https://doi.org/10.1016/j.tcs.2019.05.008