Approximation algorithms for single allocation k-hub center problem

Given a metric graph G = (V,E,w) and a positive integer k, the Single Allocation k-Hub Center problem is to find a spanning subgraph H of G such that (i) C ⊂ V is a clique of size k in H; (ii) V \ C forms an independent set in H; (iii) each v ∈ V \C is adjacent to exactly one vertex in C; and (iv) the diameter D(H) is minimized. The vertices selected in C are called hubs and the rest of vertices are called non-hubs. The Single Allocation k-Hub Center problem is NP-hard in metric graphs. In this paper, we show that for any ǫ > 0, it is NP-hard to approximate the Single Allocation k-Hub Center problem to a ratio 4 3−ǫ. Moreover, we give two approximation algorithms for solving this problem. One is a 2-approximation algorithm running in time O(n) and the other is a 53 -approximation algorithm running in time O(kn). This research is partially supported by the Ministry of Science and Technology of Taiwan under grants MOST 103–2218–E–006–019–MY3, MOST 103–2221–E– 006–135–MY3, MOST 103–2221–E–006–134–MY2. Ling-Ju Hung (corresponding author) is supported by the Ministry of Science and Technology of Taiwan under grant MOST 104–2811–E–006–056. Chia-Wei Lee is supported by the Ministry of Science and Technology of Taiwan under grant MOST 104-2811-E006-037