We propose a parallel version of the Branch & Prune (BP) algorithm for the Discretizable Distance Geometry Problem (DDGP), which consists in a subclass of Distance Geometry Problems (DGPs) that can be discretized. The main idea is to split a DDGP instance in as many subinstances as the number of processors involved in the computation, and to invoke the sequential version of BP on each processor. Due to the flexibility of the discretizing orderings that can be defined on the vertex sets of graphs representing DDGP instances, the subdivision of the original instance can be performed so that all solutions generated by locally solving the several subinstances are represented in a common coordinate system. This way, the communication phase of the parallel algorithm, where the local solutions are combined in order to generate the final set of solutions, is very efficient. We present some preliminary computational experiments and we study the behavior of the algorithm in relation to the number of considered processors. We also give some directions for transforming DDGP instances in parallelizable instances, and to modify them in order to improve the efficiency of the proposed parallel algorithm.

Abstract-Given a set of points in a Euclidean space having dimension K > 0, we are interested in the problem of finding a realization of the same set in a Euclidean space having a lower dimension. In most situations, it is not possible to preserve all available interpoint distances in the new space, so that the best possible realization, which gives the minimal error on the distances, needs to be searched. This problem is known in the scientific literature as the Multidimensional Scaling (MDS). We propose a new methodology to discretize the search space of MDS instances, with the aim of performing an efficient enumeration of their solution sets. Some preliminary computational experiments on a set of artificially generated instances are presented. We conclude our paper with some future research directions.

Palmas in the state of Tocantins is the youngest capital of Brazil and the one with the highest growth rate between 2013 and 2014 according to the Brazilian Institute of Geography and Statistics (IBGE). Presently, more than 85% of individuals live in urban centers and often need to use services related to public policies, including urban public transport. To meet this demand, the city has a company that manages and provides this service. That, however, has regularly been increasing the usage fee. These essential expenditures for the performance of services if optimized could be lower without affecting the availability and effectiveness of urban public transport. Therefore we propose the use of optimization through metaheuristics, which are algorithms that work with a certain level of randomness that throughout the process seek to find a better possible solution. Thus, this work will analyze how this problem behaves in metaheuristics applying in the scenario of Palmas, Tocantins -Brazil and discuss the optimal results expected by the algorithm, as well as identify the optimization ranges in which the metaheuristic will fit at the end of its processing.