### Abstract

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.

Original language | English |
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Pages (from-to) | 271-280 |

Number of pages | 10 |

Journal | Theoretical Computer Science |

Volume | 806 |

DOIs | |

Publication status | Published - 2020 Feb 2 |

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### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Computer Science*,

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

}

*Theoretical Computer Science*, vol. 806, pp. 271-280. https://doi.org/10.1016/j.tcs.2019.05.008

**Approximation algorithms for the p-hub center routing problem in parameterized metric graphs.** / Chen, Li Hsuan; Hsieh, Sun Yuan; Hung, Ling Ju; Klasing, Ralf.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Approximation algorithms for the p-hub center routing problem in parameterized metric graphs

AU - Chen, Li Hsuan

AU - Hsieh, Sun Yuan

AU - Hung, Ling Ju

AU - Klasing, Ralf

PY - 2020/2/2

Y1 - 2020/2/2

N2 - Let G=(V,E,w) be a Δβ-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 dH(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∈VdH(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 Δβ-metric graphs for any β>1/2. Moreover, we give 2β-approximation algorithms running in time O(n2) 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.

AB - Let G=(V,E,w) be a Δβ-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 dH(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∈VdH(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 Δβ-metric graphs for any β>1/2. Moreover, we give 2β-approximation algorithms running in time O(n2) 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.

UR - http://www.scopus.com/inward/record.url?scp=85066240262&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85066240262&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2019.05.008

DO - 10.1016/j.tcs.2019.05.008

M3 - Article

AN - SCOPUS:85066240262

VL - 806

SP - 271

EP - 280

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -