Pure 0-1 programming approaches to wireless network design

Abstract

This is a summary of the author’s Ph.D. Thesis supervised by Carlo Mannino and defended on January 18th, 2010 at Sapienza Universita di Roma. The thesis is written in English and is available from the author upon request to d.andreagiovanni@zib.de. This work deals with the Wireless Network Design Problem (WND), i.e. the problem of configuring a set of wireless transmitters in order to provide service coverage to the maximum number of receivers located in a target area. The thesis was awarded the 2010 INFORMS Doctoral Dissertation Award for Operations Research in Telecommunications, the most prestigious prize awarded in the USA to recognize outstanding scholarly achievements of young people in the field. A key task of the WND is establishing the power emitted by each transmitter. Classical optimization models represent power emissions as continuous decision variables. This modeling choice typically leads to Mixed-Integer Linear Programs with (very) ill-conditioned coefficient matrices and entails the introduction of the notorious big-M coefficients to represent disjunctive constraints that model coverage conditions. The resulting linear relaxations are very weak and the solutions returned by state-of-the-art MIP solvers are typically far from the optimum and may contain errors. The first aim of this thesis is to overcome these difficulties by introducing two innovative pure 0-1 Linear Programming formulations for the WND: the Power-Indexed Formulation (PI) and the Dyadic Formulation (DY). Both models are based on substituting the continuous power variables with the linear combination of a set of boolean variables multiplied by suitable power coefficients. Power discretization better fits a common approach to the representation of power that we have observed among practitioners. However, the only discretization is not sufficient to tackle all the numerical problems (see F. D’Andreagiovanni et al. “GUB Covers and Power-Indexed formulations for Wireless Network Design”, DIS Tech. Rep. (2) 14, Sapienza Universita di Roma, 2010): the resulting knapsack constraints still contain the coefficients that are sources of numerical issues and we thus replace them with families of (strong) valid inequalities. In particular, in the case of the PI, we define a family of lifted GUB cover inequalities, while in the case of the DY we define