On the Approximability of the Single Allocation p-Hub Center Problem with Parameterized Triangle Inequality - Chen, Li-Hsuan; Hsieh, Sun-Yuan; Hung, Ling-Ju; Klasing, Ralf (Algorithmica, 2022)
For some β ≥ 1 / 2 , a Δ β -metric graph G = ( V , E , w ) is a complete edge-weighted graph 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 vertices u , v , x ∈ V . A graph H = ( V ′ , E ′ ) is called a spanning subgraph of G = ( V , E ) if V ′ = V and E ′ ⊆ E . Given a positive integer p , let H be a spanning subgraph of G satisfying the three conditions: (i) there exists a vertex subset C ⊆ V such that C forms a clique of size p in H ; (ii) the set V \ C forms an independent set in H ; and (iii) each vertex v ∈ V \ C is adjacent to exactly one vertex in C . The vertices in C are called hubs and the vertices in V \ C are called non-hubs. The Δ β - p -Hub Center Problem ( Δ β - p HCP) is to find a spanning subgraph H of G satisfying all the three conditions such that the diameter of H is minimized. In this paper, we study Δ β - p HCP for all β ≥ 1 2 . We show that for any ϵ > 0 , to approximate Δ β - p HCP to a ratio g ( β ) - ϵ is NP-hard and we give r ( β ) -approximation algorithms for the same problem where g ( β ) and r ( β ) are functions of β . For 3 - 3 2 < β ≤ 5 + 5 10 , we give an approximation algorithm that reaches the lower bound of approximation ratio g ( β ) where g ( β ) = 3 β - 2 β 2 3 ( 1 - β ) if 3 - 3 2 < β ≤ 2 3 and g ( β ) = β + β 2 if 2 3 ≤ β ≤ 5 + 5 10 . For 5 + 5 10 ≤ β ≤ 1 , we show that g ( β ) = 4 β 2 + 3 β - 1 5 β - 1 and r ( β ) = min { β + β 2 , 4 β 2 + 5 β + 1 5 β + 1 } . Additionally, for β ≥ 1 , we show that g ( β ) = β · 4 β - 1 3 β - 1 and r ( β ) = min { β 2 + 4 β 3 , 2 β } . For β ≥ 2 , the upper bound on the approximation ratio r ( β ) = 2 β is linear in β . For 3 - 3 2 < β ≤ 5 + 5 10 , we give an approximation algorithm that reaches the lower bound of approximation ratio g ( β ) where g ( β ) = 3 β - 2 β 2 3 ( 1 - β ) if 3 - 3 2 < β ≤ 2 3 and g ( β ) = β + β 2 if 2 3 ≤ β ≤ 5 + 5 10 . For β ≤ 3 - 3 2 , we show that g ( β ) = r ( β ) = 1 , i.e., Δ β - p HCP is polynomial-time solvable.