On computing the backbone tree in large networks

In both information and public transport infrastructure construction, it is important to build a high-speed backbone tree to connect users distributed in a large area. We study the topic in this paper and model it as the inner-node weighted minimum spanning tree problem (IWMST), which asks for a spanning tree in a graph G = (V,E) (|V| = n, |E| = m) with the minimum total cost for its edges and non-leaf nodes. This problem is NP-hard because it contains the connected dominating set problem (CDS) as a special case. Since CDS cannot be approximated with a factor of (1-epsiv)H(Delta) (Delta is the maximum degree) unless NP sube DTIME[n
<sup>O(log

log

n)</sup>
] (Guha and Khuller, 1996), we can only expect a poly-logarithmic approximation algorithm for the IWMST problem. To tackle this problem, we first present a general framework for developing poly-logarithmic approximation algorithms. Our framework aims to find a k/k-1 ln n-approximate algorithm (k isin N and k ges 2) for the IWMST problem. Based on this framework, we further design a polynomial time approximation algorithms which can find a 2 ln n-approximate solution in O(mn log n) time.