GUB Covers and Power-Indexed Formulations for Wireless Network Design

Wireless networks have shown a rapid growth over the past two decades and now play a key role in new generation telecommunications networks. The physical medium of wireless networks is the radio spectrum, a scarce resource which is becoming extremely congested and needs to be allocated in more effective ways. Since the early 1980s several optimization models have been developed to design wireless networks. In this paper we propose a pure 0-1 formulation which is able to model a very general situation in which both emission powers and operating frequencies can be optimized. In contrast with the classical mixed integer formulation, where powers are represented by continuous variables, we consider only a finite set of transmitting powers. The ensuing model has two major advantages: it better fits the usual practice and minimizes the numerical problems produced by the interaction of continuous and 0-1 decision variables. A crucial ingredient of our approach is an effective basic formulation for the single knapsack problem representing the coverage condition of a receiver. This formulation is based on the well-known lifted GUB cover inequalities introduced by Wolsey and its core is a slight extension of the exact formulation proposed by Wolsey for the GUB knapsack polytope with two GUB constraints. In the specific framework of our real-life problem the two GUB constraints case corresponds to the very common situation in which only one major interferer is present. The effectiveness of such formulation is assessed by comprehensive computational results.