Abstract
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.
| Original language | English |
|---|---|
| Journal | Theoretical Computer Science |
| DOIs | |
| Publication status | Published - 2019 Jan 1 |
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All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Computer Science(all)
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Approximation algorithms for the p-hub center routing problem in parameterized metric graphs. / Chen, Li Hsuan; Hsieh, Sun-Yuan; Hung, Ling Ju; Klasing, Ralf.
In: Theoretical Computer Science, 01.01.2019.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 - 2019/1/1
Y1 - 2019/1/1
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
JO - Theoretical Computer Science
JF - Theoretical Computer Science
SN - 0304-3975
ER -