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

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∈Vd_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(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.