On Min-Power Steiner Tree
  In the classical (min-cost) Steiner tree problem, we are given an edge-weighted undirected graph and a set of terminal nodes. The goal is to compute a min-cost tree S which spans all terminals. In this paper we consider the min-power version of the problem, which is better suited for wireless applications. Here, the goal is to minimize the total power consumption of nodes, where the power of a node v is the maximum cost of any edge of S incident to v. Intuitively, nodes are antennas (part of which are terminals that we need to connect) and edge costs define the power to connect their endpoints via bidirectional links (so as to support protocols with ack messages). Differently from its min-cost counterpart, min-power Steiner tree is NP-hard even in the spanning tree case, i.e. when all nodes are terminals. Since the power of any tree is within once and twice its cost, computing a rho \leq ln(4)+eps [Byrka et al.'10] approximate min-cost Steiner tree provides a 2rho<2.78 approximation for the problem. For min-power spanning tree the same approach provides a 2 approximation, which was improved to 5/3+eps with a non-trivial approach in [Althaus et al.'06]. Here we present an improved approximation algorithm for min-power Steiner tree. Our result is based on two main ingredients. We prove the first decomposition theorem for min-power Steiner tree, in the spirit of analogous structural results for min-cost Steiner tree and min-power spanning tree. Based on this theorem, we define a proper LP relaxation, that we exploit within the iterative randomized rounding framework in [Byrka et al.'10]. A careful analysis provides a 3ln 4-9/4+eps<1.91 approximation factor. The same approach gives an improved 1.5+eps approximation for min-power spanning tree as well, matching the approximation factor in [Nutov and Yaroshevitch'09] for the special case of min-power spanning tree with edge weights in {0,1}. 
Fabrizio Grandoni
cs.DS