1 Shortest Path Algorithms in Transportation models :

Shortest Path Problems are among the most studied network flow optimization problems, with interesting applications in various fields. One such field is transportation, where shortest path problems of different kinds need to be solved. Due to the nature of the application, transportation scientists need very flexible and efficient shortest path procedures, both from the running time point of view, and also for the memory requirements. Since no “best” algorithm currently exists for every kind of transportation problem, research in this field has recently moved to the design and implementation of “ad hoc” shortest path procedures, which are able to capture the peculiarities of the problems under consideration. The aim of this work is to present in a unifying framework both the main algorithmic approaches that have been proposed in the past years for solving the shortest path problems arising most frequently in the transportation field, and also some important implementation techniques which allow one to derive efficient procedures from the general algorithmic schema, in line with trends in current research. More precisely, in the first part of the paper, after presenting the problem, we briefly review those classical primal and dual algorithms which seem to be the most interesting in transportation, either as a result of theoretical considerations or because of their efficiency, and also in view of their practical use in transportation models. Promising reoptimization approaches are then discussed. The second part is devoted to dynamic shortest path problems, that arise very frequently in the transportation field. We analyze the main features of the problems and present, under suitable conditions on travel time and cost functions, a general “chronological” algorithmic paradigm, called Chrono-SPT. We show the potentiality of this paradigm, and, by properly performing some implementation choices, we study several, interesting special cases. Finally, we investigate some promising research avenues related to various extensions. These include “timewindows” problems, “turn penalties and prohibitions”, and “multicriteria” problems, which often can be solved by extending the ideas discussed for the dynamic case. Finally, “shortest hyperpaths” problems are briefly introduced, due to their practical relevance in many transportation applications. 1 Research grants by C.N.R. Progetto Finalizzato Trasporti II, 74.9600089 and by C.N.R. G.N.I.M.