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) (vertical bar V vertical bar = n. vertical bar E vertical bar = 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 E)H(A) (A is the maximum degree) unless NP subset of DT I M E/n(O(log log n)) vertical bar [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 r. rk Inn-approximate Algorithm (k epsilon 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 k/k-1 In n-approximate solution in O(mn log n) time, while the second one can compute a 1.5 Inn-approximate solution in O(n(2) Delta(6)) time (Delta is the maximum degree in G). We have also studied the relationships between the IWMST problem and several other similar problems.