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

• Published in: Theoretical computer science Vol. 806; pp. 271 - 280
• Main Authors: Chen, Li-Hsuan, Hsieh, Sun-Yuan, Hung, Ling-Ju, Klasing, Ralf
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 02.02.2020; Elsevier
• Subjects: Computer Science; Hub allocation; Metric graphs; Networking and Internet Architecture; Stability of approximation; β-Triangle inequality; Stability of approximation; Metric graphs; Hub allocation; β-Triangle inequality
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2019.05.008

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 mostp-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 mostp-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.