The centroid as an estimate for the quadratic min-power centre

Given a set of nodes in the plane, the min-power centre is a point that minimises the cost of the star centred at this point and spanning all nodes. The cost of the star is defined as the sum of the costs of its nodes, where the cost of a node is an increasing function of the length of its longest incident edge. The min-power centre problem provides a model for optimally locating a cluster-head amongst a set of radio transmitters. We provide upper bounds for the performance of the centroid (centre of mass) of the given nodes as an approximation to the quadratic min-power centre