Approximating Spanning Tree with Weighted Inner Nodes

We consider a problem coming from practical applications: finding a minimum spanning tree with both edge weights and inner node (non-leaf node) weights. This problem is NP-complete even in the metric space. We present two polynomial time algorithms which achieve approximation factors of 2.35 ln n and 2(Hn − 1), respectively, where n is the number of nodes in the graph and Hn is the n-th Harmonic number. This nearly matches the lower bound: no polynomial-time approximation algorithm can achieve an approximation factor of (1−2)Hn, for any 2 > 0. For metric case where the edge weights are symmetric and satisfy the triangle inequality, it also proves to be an NP-hard problem and we give a 3.52 approximation algorithm and an improved factor 3.106 one and also show that an approximation factor of 1.463 is impossible unless NP ⊆ DTIME[n]. We also give an approximation algorithm with factor ∆−1, where ∆ is the maximum degree of the graph.