Approximability and inapproximability of the star p -hub center problem with parameterized triangle inequality
Résumé
A complete weighted graph G = (V, E, w) is called ∆ β-metric, 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) and a center c ∈ V , and an integer p, the ∆ β-Star p-Hub Center problem (∆ β-SpHCP) is to find a depth-2 spanning tree T of G rooted at c such that c has exactly p children (also called hubs) and the diameter of T is minimized. In this paper, we study ∆ β-SpHCP for all β ≥ 1 2. We show that for any ǫ > 0, to approximate the ∆ β-SpHCP to a ratio g(β) − ǫ is NP-hard and give r(β)-approximation algorithms for the same problem where g(β) and r(β) are functions of β. A subclass of metric graphs is identified that ∆ β-SpHCP is polynomial-time solvable. Moreover, some r(β)-approximation algorithms given in this paper meet approximation lower bounds.
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