Approximation algorithms for inner-node weighted minimum spanning trees

This paper addresses 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 (I - ε)H(Δ) (Δ is the maximum degree) unless NP ⊆ DTIME|n ^{O(log log n)}] [10], 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 two polynomial time approximation algorithms. The first one can find a 2 In n-approximate solution in O(mn log n) time, while the second one can compute a 1.5 In n-approximate solution in O(n²Δ⁶) time (A is the maximum degree in C). We have also studied the relationships between the IWMST problem and several other similar problems.