Shortest Path Algorithms In Transportation Models: Classical and Innovative Aspects

Abstract

Shortest Path problems are among the most studied network flow optimization problems. Since the end of the 1950's, more than two thousand scientific works have been published in the literature, most of them in journals and conference proceedings concerning general combinatorial optimization on graphs, but also in numerous specialized journals. One of the most interesting application fields is transportation.In many transportation problems, shortest path problems of different kinds need to be solved. These include both classical problems, for example to determine shortest paths (under various measures, such as length, cost and so on) between some given origin/destination pairs in a certain area, and also non standard versions, for example to compute shortest paths either under additional constraints or on particular structured graphs. Due to the nature of the applications, transportation scientists need very flexible and efficient shortest path procedures, both from the running time point of view and in terms of memory requirements. Since no "best" algorithm exists for every kind of transportation problem, Le. no algorithm exists which shows the same practical behavior independently of the structure of the graph, of its size and of the cost measure used for evaluating the paths, research in this field has recently moved to the design and the implementation of "ad hoc" shortest path procedures, which are able to capture the peculiarities of the problems under consideration. Much of the focus has been on the choice and implementation of efficient data structures (see, for instance, Tarjan, 1983). P. Marcotte et al. (eds.), Equilibrium and Advanced Transportation Modelling

State-based accelerations and bidirectional search for bi-objective multi-modal shortest paths

“…The goal is to find the non-dominated viable shortest paths considering the minimization of the travel time and of the number of modal transfers. We show that the problem, initially considered by Lozano and Storchi [15], is a polynomially-solvable bi-objective variant of the mono-objective regular language-constrained shortest path problem [2,8].We propose several label setting algorithms for solving the problem: a topological label-setting algorithm improving on algorithms proposed by Pallottino and Scutellà [23] and Lozano and Storchi [15], a multi-label algorithm using buckets and its bidirectional variant, as well as dedicated goal oriented techniques. Furthermore, we propose a new NFA state-based dominance rule.…”

“…We propose several label setting algorithms for solving the problem: a topological label-setting algorithm improving on algorithms proposed by Pallottino and Scutellà [23] and Lozano and Storchi [15], a multi-label algorithm using buckets and its bidirectional variant, as well as dedicated goal oriented techniques. Furthermore, we propose a new NFA state-based dominance rule.…”

Solution approaches for determining user-oriented paths on dynamic networks

“…The same authors (Orda and Rom, 1991) demonstrate that the optimality principles, which represent the theoretical fundamentals of labelling procedures, are not verified when the cost function does not represent time. Pallottino and Scutellà (1997) address the TDSPP defined on non-FIFO graphs. The waiting at nodes is allowed with a penalty cost for each waiting time instant and the cost function is assumed to be discrete.…”

“…The waiting at nodes is allowed with a penalty cost for each waiting time instant and the cost function is assumed to be discrete. The label-correcting algorithm, proposed by Pallottino and Scutellà (1997), implicitly works on an expanse graph. In particular, each node has as many copies as the considered time instants in the time horizon.…”

“…In general, the scientific literature refers to two main structures of dynamic graphs: FIFO and non-FIFO. A graph is said to be FIFO when, given two users u 1 and u 2 , if u 1 passes through an arc starting from the instant time t 1 and u 2 from t 2 > t 1 , then u 1 must arrive at tail node before user u 2 , see e.g., Pallottino and Scutellà (1997).…”