Symmetric connectivity with minimum power consumption in radio networks

We study the problem of assigning transmission ranges to the nodes of a multi-hop packet radio network (also known as static ad hoc wireless network) so as to minimize the total power consumed under the constraint that enough power is provided to the nodes to ensure that the network is connected. Precisely, we require that the bidirectional links established by the transmission range of every node form a connected graph. We call this problem MIN-POWER SYMMETRIC CONNECTIVITY. Implicit in previous work on transmission range assignment under asymmetric connectivity requirements is the proof that MIN-POWER SYMMETRIC CONNECTIVITY is NP-hard and that a simple algorithm has approximation ratio of 2. In this paper we establish the similarity between MIN-POWER SYMMETRIC CONNECTIVITY and the classic STEINER TREE problem in graphs, give a polynomial-time approximation scheme with performance ratio approaching 1 + ln 2 approximate to 1.69, and present a more practical 15/8 approximation algorithm. We also show that the related MIN-POWER SYMMETRIC UNICAST problem can be solved efficiently by a shortest-path computation in an appropriately constructed graph.