Gamma Deployment Problem in Grids: Complexity and a new Integer Linear Programming Formulation - Marcelo Fonseca Faraj (2019)
Vehicular ad hoc networks are one of the most significant components of intelligent transportation systems. They have the potential to ease traffic management, lower accident rates and provide other solutions to smart cities. One of the main challenges on vehicular ad hoc networks is to choose the best places to deploy roadside units. This thesis deals with the Gamma Deployment Problem, which consists of deploying the minimum number of roadside units on a road network meeting the Gamma Deployment metric. Within this metric, at least a given fraction of vehicles passing in the road network must be covered, i.e they should meet at least one roadside unit each predetermined time interval. In this thesis, I propose a formal treatment based on graph theoretical concepts and provide a proof that the decision version of the Gamma Deployment Problem in Grids is NP-complete. In addition, I expose an issue with the multi-flow integer linear programming formulation present in literature and propose a slight correction for it. I also introduce a new integer linear programming formulation based on set covering and provide a proof that the polytope associated with its linear programming relaxation is contained in the polytope associated with the linear programming relaxation of the multi-flow formulation. Finally, computational experiments with a commercial optimizer show that the set covering formulation widely outperforms the multi-flow formulation regarding linear programming relaxation gap and running time.