## 2 Overview of Multiobjective Path Problems and Algorithms
### Types of Multiobjective Path Problems
#### 2.2.1.1 Apo--A Posteriori Aggregation of Preferences Methods  
1. Maximal Complete Set Computation Labeling techniques \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Reference** & **\# Objectives** & **Technique \({}^{*}\)** & **Class** \\ \hline ([PERSON] 1980) [8] & 2 & ls & APO \\ ([PERSON] and [PERSON] 1982) [11] & 2 & ran & APO \\ ([PERSON] 1984) [12] & \(k\) & ls & APO \\ ([PERSON] and [PERSON] 1985) [13] & \(k\) & lc & APO \\ ([PERSON] et al., 1991) [14] & 2 & 2p & APO \\ ([PERSON] and [PERSON] 1991) [15] & \(k\) & ls & APO \\ ([PERSON] and [PERSON], 1992) [16] & \(k\) & ls & APO \\ ([PERSON] 1999) [9] & \(k\) & ls/lc & APO \\ ([PERSON] and [PERSON] 2001) [17] & \(k\) & ls/lc & APO \\ ([PERSON] et al., 2003) [18] & 2 & ran & APO \\ ([PERSON] and [PERSON] 2010) [19] & 2 & lc & APO \\ ([PERSON] et al., 2012) [20] & 2 & ls & APO \\ ([PERSON] and [PERSON] 2012) [21] & \(k\) & par & APO \\ ([PERSON] et al., 2013) [22] & \(k\) & ls & APO \\ ([PERSON] and [PERSON] 2013) [23] & \(k\) & ls & APO \\ ([PERSON] et al., 2015) [24] & 2 & rec & APO \\ ([PERSON] et al., 2015) [25] & \(k\) & ls & APO \\ ([PERSON] and [PERSON] 2016) [26] & 2 & ls & APO \\ ([PERSON] et al., 2016) [27] & 2 & ls & APO \\ ([PERSON] and [PERSON] 2019) [28] & 2 & ls & APO \\ ([PERSON] et al., 2021) [29] & \(k\) & ls & APO \\ ([PERSON] et al., 2021) [30] & \(k\) & ls & APO \\ ([PERSON] et al., 2022) [31] & \(k\) & lc & APO \\ ([PERSON] et al., 2023) [32] & \(k\) & ls & APO \\ ([PERSON] et al., 2023) [33] & 2 & ls & APO \\ ([PERSON] et al., 2023) [34] & \(k\) & ls & APO \\ ([PERSON] and [PERSON] 2023) [35] & 2 & ls & APO \\ \hline ([PERSON] et al., 1990) [36] & 2 & 2p & INT \\ ([PERSON] and [PERSON] 1994) [37] & 2 & 2p & INT \\ ([PERSON] 1994) [38] & 2 & lc & INT \\ ([PERSON] et al., 1999) [39] & 2 & 2p/ran & INT \\ \hline ([PERSON] et al., 2003) [40] & \(k\) & lc/ran & APR \\ ([PERSON] et al., 2006) [41] & 2 & lc/ran & APR \\ ([PERSON] and [PERSON] 2010) [42] & \(k\) & lc & APR \\ ([PERSON] et al., 2011) [43] & \(k\) & ls & APR \\ ([PERSON] et al., 2014) [44] & \(k\) & ls & APR \\ ([PERSON] and [PERSON] 2018) [45] & \(k\) & ls & APR \\ ([PERSON] et al., 2020) [46] & \(k\) & 2p/lc & APR \\ \hline \hline \end{tabular} \({}^{*}\) ls: label setting; lc: label correcting; par: parametric; ran: ranking; rec: recursive; 2p: two phases.  
\end{table} 
Table 1: Summary of some essential features of exact algorithms for the MOPPs with additive metrics. Labeling algorithms for the MOSPP are generalizations of the single objective labeling methods. Since we have now more than one objective function the algorithm has to consider more than one path starting in \(s\) up to each node, implying that more than one label associated with that node is used. Note that the labels associated with one node may dominate one another. This type of algorithm is founded on an adaptation of the Principle of Optimality valid for the shortest path problem that states that every efficient path is formed by efficient subpaths. This can be stated if no cycles with a negative cost exist in the network representation. Like in the single objective problem, these algorithms may be grouped into two types: _label setting_ or _label correcting_ algorithms. The latter type can still be divided into those algorithms that make a label correspond to a path starting in \(s\) (these are designated as label-selection methods) or those that are characterized by associating a label with each node and representing several paths from \(s\) to that node so that whenever a label is chosen all the associated paths are expanded by using the network arcs (these are designated as node-selection methods).