### abstract

- In this paper, we present approximation algorithms for a variety of problems occurring in the design of energy-efficient wireless communication networks. We first study the k-station network problem, where for a set S of stations and some constant k, one wants to assign transmission powers to at most k senders such that every station in S can receive a signal from at least one sender. We give a (1 + ϵ)-approximation algorithm for this problem. The second problem deals with energy-efficient networks, allowing bounded hop multicast operations, that is given a subset C of the stations S and a designated source node s ∈ S, we want to assign powers to the sending stations, such that every node in C can be reached by a transmission from s within k hops. For this problem, we provide an algorithm which runs in time linear in ∣S∣. The last problem deals with a variant of the non-metric TSP problem where the edge costs correspond to the Euclidean distances to the power of some α ⩾ 1; this problem is motivated by data aggregation schemes in wireless sensor networks. We provide a simple constant approximation algorithm, which improves upon previous results when 2 ⩽ α ⩽ 2.7.