## 2 Overview of Multiobjective Path Problems and Algorithms
### Types of Multiobjective Path Problems
#### 2.2.1.1 Apo--A Posteriori Aggregation of Preferences Methods
For two or three objectives, the label-correcting implementation using a first in-first out (FIFO) approach outperformed both label setting and the deletion ranking method in [66]. However, for a small set of tests, a similar implementation using the algorithm in [71] was faster than the remaining ones. Paper [18] deals with a BOSPP with additional constraints, which imposes that paths cannot exceed a maximal bound concerning an additive metric. The work reports experiments where the label setting method is faster than the ranking method using the algorithm in [71]. However, deviation algorithms are very flexible whenever additional constraints are imposed because the non-feasible paths can be discarded at an early phase. The ranking method in [64] may be used to solve the MOSPP. In [11], it was used for a particular case with two objectives. Two-phase techniques  
These algorithms work in two phases, analogously to what is described in the context of interactive methods for the MOSPP in [36] and in a general biobjective optimization problem context in [76]. In phase 1 of the algorithm, extreme, supported efficient solutions are calculated very often by solving single-objective weighted shortest path problems. In phase 2, the remaining supported and non-supported efficient solutions are calculated by enumeration. Note that, in phase 2, the search space of the solutions is restricted to the duality gaps. Therefore, those problems can be solved faster than by applying pure enumeration to the BOSPP. The unimodularity property of the constraints is used by [14] in the considered linear programming formulation of the BOSPP. The proposed method solves a parametric shortest path problem for finding the efficient supported solutions. The unsupported solutions are obtained with a label correction method. This approach is illustrated in Figure 5.  
Recursive methods  
An exact recursive algorithm (considering non-negative arc costs) is proposed in [24], using implicit enumeration based on the 'pulse algorithm' for the constrained shortest path problem given in [63]. Different types of pruning are applied: pruning paths with cycles, pruning based on a dominance test, pruning Nadir points based on upper bounds for each objective function, with both techniques being used in complete and partial paths. These procedures are particularly suited for implicit enumeration. This method is also extended to more than two OFs. Minimal complete set computation  
Labeling techniques  
A generalization of [PERSON]'s algorithm to the biobjective shortest path problem is presented in [28], by keeping only one candidate label per node in a priority queue of size \(n\). Thence, a novel algorithm for the one node-to-all nodes BOSPP, enabling the determination of all the nondominated points in the objective function space and one efficient path associated with each of these points, is introduced. For the one-to-one BOSPP, the classical bidirectional search scheme is incorporated into the procedure. The proposed procedure also involves pruning strategies so that the computation of unnecessary labels is avoided.