Labeling Methods for Partially Ordered Paths

TL;DR

The paper introduces the partial order shortest path problem (POSP), a generalization of MOSP where path weights live in a partly ordered space $(\mathcal{T},\preceq)$. It identifies structural conditions—history-freeness, (weak) independence, arc-/cycle-related properties, and $\mu$-boundedness—that guarantee optimal substructure and enable two labeling approaches: label-correcting and label-setting. By leveraging the Multi-Objective Dijkstra Algorithm (MDA) as a POSP backbone, the authors achieve improved, output-sensitive runtime bounds and demonstrate applicability across diverse variants (time-dependent, interval weights, bottleneck, EV charging, replenishment) on general digraphs. A comprehensive lookup table is provided to guide practitioners in selecting the appropriate algorithm for a given weight structure. Overall, the work unifies ad hoc POSP methods under a principled framework, extending MOSP techniques to a broad class of partial-orders with practical implications for complex, multi-criteria path problems.

Abstract

The landscape of applications and subroutines relying on shortest path computations continues to grow steadily. This growth is driven by the undeniable success of shortest path algorithms in theory and practice. It also introduces new challenges as the models and assessing the optimality of paths become more complicated. Hence, multiple recent publications in the field adapt existing labeling methods in an ad hoc fashion to their specific problem variant without considering the underlying general structure: they always deal with multi-criteria scenarios, and those criteria define different partial orders on the paths. In this paper, we introduce the partial order shortest path problem (POSP), a generalization of the multi-objective shortest path problem (MOSP) and in turn also of the classical shortest path problem. POSP captures the particular structure of many shortest path applications as special cases. In this generality, we study optimality conditions or the lack of them, depending on the objective functions' properties. Our final contribution is a big lookup table summarizing our findings and providing the reader with an easy way to choose among the most recent multi-criteria shortest path algorithms depending on their problems' weight structure. Examples range from time-dependent shortest path and bottleneck path problems to the electric vehicle shortest path problem with recharging and complex financial weight functions studied in the public transportation community. Our results hold for general digraphs and, therefore, surpass previous generalizations that were limited to acyclic graphs.

Figures (2)

• Figure 1: The graph and weight space used in \ref{['ex:SPO_NOT_SUFFICIENT']}.
• Figure 2: Consider the instance $(G,c,\mathbb{N},\leq,s)$ given by the graph above, in which each path $p$ starting in $s$ is annotated with a weight $c(p)$ drawn from $\mathbb{N}$. The partial order $\leq$ is the natural total order on $\mathbb{N}$. It is easy to see that $(G,c,\mathbb{N},\leq,s)$ is well-posed, history-free, weakly subpath optimal, and even arc-increasing but not weakly independent.

Theorems & Definitions (52)

• Definition 2.1: $\text{POSP}$
• Definition 2.2: $MOSP$
• Definition 2.3
• Definition 2.4: $\mu$-Bounded
• Definition 3.1: History-Free
• Definition 3.2: Subpath Optimality
• Definition 3.3: Independence Perny2005
• Example 3.1
• Proposition 3.1
• proof