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 Innernode 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-ε)H(Δ) (Δ is the maximum degree) unless NP C⊆ DTIME[n^{O(log log n)}] [7], 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 In n-approximate Algorithm (k ∈ N and k ≥ 2) for the IWMST problem. Based on this framework, we further design a polynomial time approximation algorithms which can find a 2 In n-approximate solution in O(mn log n) time.