On the bounded-hop range assignment problem

We study the problem of assigning transmission ranges to radio stations in the plane such that any pair of stations can communicate within a bounded number of hops h and the cost of the network is minimized. The cost of transmitting in a range r is proportional to r^α, where α ≥ 1. We consider two settings of this problem: collinear station locations and arbitrary locations. For the case of collinear stations, we introduce the pioneer polynomial-time exact algorithm for any α ≥ 1 and constant h, and thus conclude that the 1D version of the problem, where h is a constant, is in P. For an arbitrary h, not necessarily a constant, and α = 1, we propose a 1.5-approximation algorithm. This improves the previously best known approximation ratio of 2. For the case of stations placed arbitrarily in the plane, we present a (6 + ε)-approximation algorithm, for any ε > 0. This improves the previously best known approximation ratio of 4(9^h−2)/(^h√ 2−1). Moreover, we show a (1.5+ε)-approximation algorithm for a case where deviation of one hop (h + 1 hops in total) is acceptable.