The Minimum Power Broadcast problem in Wireless Networks: a Simulated Annealing approach Roberto Montemanni, Luca Maria Gambardella Arindam K. Das Istituto Dalle Molle di Studi sull’Intelligenza Artificiale (IDSIA), Department of Electrical Engineering, University of Washington, Galleria 2, CH-6928 Manno-Lugano, Switzerland. Box 352500, Seattle, WA 98195, U.S.A. Email addresses: {roberto, luca}@idsia.ch Email address: arindam@ee.washington.edu Abstract— Broadcasting in wireless networks, unlike wired networks, inherently reaches several nodes with a single trans- mission. For omnidirectional wireless broadcast to a node, all nodes closer will also be reached. This property can be used to compute routing trees which minimize the sum of the transmitter powers. In this paper we present a mixed integer programming for- mulation and a simulated annealing algorithm for the problem. Extensive experimental results for the heuristic approach are presented. They show that the algorithm we propose is capable of improving the results of state-of-the-art algorithms for most of the problems considered. The solutions provided by the simulated annealing algorithm can be improved by applying a very fast post-optimization procedure. This leads to the best known mean results for the problems considered. I. I NTRODUCTION Among the most crucial issues related to ad-hoc and sensor networks is that of operation in limited energy environments, since devices are usually equipped with battery with a limited lifetime. Since radio signals have non-linear attenuation properties, it is very energy-consuming to transmit a signal far away. Another drawback of long-range transmissions is that they tend to produce widespread interference over the network, and for this reason they should be avoided. The previous issues can be seen as correlated, and they can be handled together by taking advantage of the so-called wireless multicast advantage property (see, Wieselthier et al. [1]). This property is based on the observation that, in wireless networks, devices are usually equipped with omnidirectional antennae, and for this reason multiple nodes can be reached by a single transmission. In the example of Figure 1a, nodes j and k are closer to node i than node m, then the signal originating in node i, and directed to node m, will be received also by nodes j and k, since they are within the transmission range of a communication from node i to node m. For a given network with an identified source node, the MPB (minimum power broadcast) problem is to assign trans- mission powers to the nodes in such a way that the network is connected and the total power consumption is minimized. The MPB problem in wireless networks is shown to be NP- complete in Cagalj et al. [2], and this implies that polynomial time algorithms are unlikely to exist. Some mixed integer Fig. 1. (a) Communication model. (b) Costs for mathematical formulation MIP . c ij is the power required to reach j from i, while c ik is the additional power required to reach k when j is already reached from i. Analogously, c im is the additional power required to reach node m from i while k is already reached. programming formulations for the problem are described in Das et al. [3]. Wieselthier et al. [1] first observed that the so called “node based” approach is more suitable for wireless environment than the previously adopted “link-based” algorithms. They developed the Broadcast Incremental Power (BIP) algorithm, which is a simple sub-optimal heuristic for constructing min- imum power broadcast trees in wireless networks. In this algorithm, new nodes are added to the tree on a minimum incremental cost basis, until all intended destination nodes are included. It was subsequently shown in Wan et al. [4] that the BIP algorithm has an approximation ratio between 13/3 and 12. Other techniques that have been suggested for solving this problem include an internal nodes based broadcasting produce by Stojmenovic et al. [5], an evolutionary approach by Mark et al. [6], a localized algorithm by Cartigny et al. [7], a swarm based procedure by Das et al. [8] (Ant Colony System, ACS, see also Gambardella and Dorigo [9]). This last algorithm was hybridized within a cluster-merge (CM) method presented in Das et al. [10]. Some heuristic approaches for improving solutions provided by other methods was presented in Das et al. [11]. Most of these heuristic techniques are described in detail in Das et al. [12]. The rest of the paper is organized as follows. In Section II, we outline the network model. A mixed integer programming formulation, based on a novel (for the problem) incremental cost mechanism, is described in Section III. The simulated annealing paradigm and its adaptation to the MPB problem is presented in Section IV. Computational results are presented