Average Case Analysis of the MST-heuristic for the Power Assignment Problem: Special Cases

We present an average case analysis of the minimum spanning tree heuristic for the power assignment problem. The worst-case approximation ratio of this heuristic is 2. We have the following results: (a) In the one-dimensional case, with uniform [0, 1]-distributed distances, the expected approximation ratio is bounded above by 2− 2/(p+ 2), where p denotes the distance power gradient. (b) For the complete graph, with uniform [0, 1] distributed edge weights, the expected approximation ratio is bounded above by 2−1/2ζ(3), where ζ denotes the Riemann zeta function.