On the Complexity of the Star p-hub Center Problem with Parameterized Triangle Inequality.

A complete weighted graph G = (V, E, w) is called Delta(beta)-metric, for some beta >= 1/2, if G satisfies the beta-triangle inequality, i.e., w(u, v) = beta center dot(w(u, x) + w(x, v)) for all vertices u, v, x is an element of V. Given a Delta(beta)-metric graph G = (V, E, w) and a center c is an element of V, and an integer p, the Delta(beta)-Star p-Hub Center Problem (Delta(beta)-SpHCP) is to find a depth-2 spanning tree T of G rooted at c such that c has exactly p children and the diameter of T is minimized. The children of c in T are called hubs. For beta = 1, Delta(beta)-SpHCP is NP-hard. (Chen et al., COCOON 2016) proved that for any epsilon > 0, it is NP-hard to approximate the Delta(beta)-SpHCP to within a ratio 1.5 - epsilon for beta = 1. In the same paper, a 5/3-approximation algorithm was given for Delta(beta)-SpHCP for beta = 1. In this paper, we study Delta(beta)-SpHCP for all beta >= 1/2. We show that for any epsilon > 0, to approximate the Delta(beta)-SpHCP to a ratio g(beta) - epsilon is NP-hard and we give r(beta)-approximation algorithms for the same problem where g(beta) and r(beta) are functions of beta. If beta <= 3-root 3/2, we have r(beta) = g(beta) = 1, i.e., Delta(beta)-SpHCP is polynomial time solvable. If 3-root 3/2 < beta <= 2/3, we have r(beta) = g(beta) = 1+2 beta-2 beta(2)/4(1-beta). For 2/3 <= beta <= 1, r(beta) = min{1+2 beta-2 beta(2)/4(1-beta),1 + 4 beta(2)/5 beta+1}. Moreover, for beta >= 1, we have r(beta) = min{beta + 4 beta(2)-2 beta/2+beta, 2 beta+1}. For beta >= 2, the approximability of the problem (i.e., upper and lower bound) is linear in beta.