Label Correcting Algorithms for the Multiobjective Temporal Shortest Path Problem

TL;DR

This paper develops label correcting algorithms for multiobjective temporal shortest path problems without assuming monotonicity or isotonicity, addressing challenges posed by zero-duration cycles and path length restrictions.

Contribution

It introduces algorithms for the problem without monotonicity/isotonicity assumptions and identifies conditions where path length bounds are unnecessary.

Findings

Algorithms successfully compute all nondominated images under certain conditions.

Study extends shortest path methods to non-monotonic, non-isotonic temporal graphs.

Addresses traversal of zero-duration cycles with path length restrictions.

Abstract

Given a directed, discrete-time temporal graph $G=(V,R)$, a start node $s\in V$, and $p\geq1$ objectives, the single-source multiobjective temporal shortest path problem asks, for each $v\in V$, for the set of nondominated images of temporal $s$-$v$-paths together with a corresponding efficient path for each image. A recent general label setting algorithm for this problem relies on two properties of the objectives - monotonicity and isotonicity. Monotonicity generalizes the nonnegativity assumption required by label setting methods for the classical additive single-objective shortest path problem on static graphs, while isotonicity ensures that the order of the objective values of two paths is preserved when both are extended by the same arc. In this paper, we study the problem without assuming monotonicity and/or isotonicity. A key difficulty in this setting is that zero-duration temporal cycles may need to be traversed an arbitrary finite number of times to generate all nondominated images. This motivates the study of a restricted problem variant in which a maximum admissible path length $K$ is imposed, and only paths containing at most $K$ arcs are considered. We develop general label correcting algorithms for this setting and establish several sufficient conditions under which such a bound is not required, implying that the algorithms compute all nondominated images.