On the power assignment problem in radio networks

Given a finite set S of points (i.e. the stations of a radio network) on a d-dimensional Euclidean space and a positive integer 1 ≤ h ≤ |S| - 1, the Min dD h-Range Assignment problem consists of assigning transmission ranges to the stations so as to minimize the total power consumption, provided that the transmission ranges of the stations ensure the communication between any pair of stations in at most h hops. Two main issues related to this problem are considered in this paper: the trade-off between the power consumption and the number of hops; the computational complexity of the Min dD-h-Range Assignment problem. As for the first question, we provide a lower bound on the minimum power consumption of stations on the plane for constant h. The lower bound is a function of |S|, h and the minimum distance over all the pairs of stations in S. Then, we derive a constructive upper bound as a function of |S|, h and the maximum distance over all pairs of stations in S (i.e. the diameter of S). It turns out that when the minimum distance between any two stations is "not too small" (i.e. well spread instances) the upper bound matches the lower bound. Previous results for this problem were known only for very special 1-dimensional configurations (i.e., when points are arranged on a line at unitary distance) [Kirousis, Kranakis, Krizanc and Pelc, 1997]. As for the second question, we observe that the tightness of our upper bound implies that Min 2D h-Range Assignment restricted to well spread instances admits a polynomial time approximation algorithm. Then, we also show that the same approximation result can be obtained for random instances. On the other hand, we prove that for h = |S| - 1 (i.e. the unbounded case) Min 2D h-Range Assignment is NP-hard and Min 3D h-Range Assignment is APX-complete.