Approximation algorithms for the p-hub center routing problem in parameterized metric graphs
Résumé
A complete weighted graph G = (V, E, w) is called ∆ βmetric, for some β ≥ 1/2, if G satisfies the β-triangle inequality, i.e., w(u, v) ≤ β • (w(u, x) + w(x, v)) for all vertices u, v, x ∈ V. Given a ∆ β-metric graph G = (V, E, w), the Single Allocation at most p-Hub Center Routing problem is to find a spanning subgraph H * of G such that (i) any pair of vertices in C * is adjacent in H * where C * ⊂ V and |C * | ≤ p; (ii) any pair of vertices in V \ C * is not adjacent in H * ; (iii) each v ∈ V \ C * is adjacent to exactly one vertex in C * ; and (iv) the routing cost r(H *) = u,v∈V dH * (u, v) is minimized where dH * (u, v) = w(u, f * (u)) + w(f * (u), f * (v)) + w(v, f * (v)) and f * (u), f * (v) are the vertices in C * adjacent to u and v in H * , respectively. Note that w(v, f * (v)) = 0 if v ∈ C *. The vertices selected in C * are called hubs and the rest of vertices are called non-hubs. In this paper, we show that the Single Allocation at most p-Hub Center Routing problem is NP-hard in ∆ β-metric graphs for any β > 1/2. Moreover, we give 2βapproximation algorithms running in time O(n 2) for any β > 1/2 where n is the number of vertices in the input graph.
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