Label Correcting Algorithms for the Multiobjective Temporal Shortest Path Problem

2026-05-07

AI Summary

This work addresses the single-source multi-objective shortest path problem on directed discrete-time temporal graphs where objective functions may violate monotonicity or isotonicity assumptions. The authors propose a general label-correcting algorithm that efficiently computes the set of non-dominated images without relying on restrictive properties of the objective functions, thereby overcoming a key limitation of traditional label-setting approaches. To handle temporal cycles with zero duration, the method incorporates an upper bound \(K\) on path length; however, under certain conditions, it guarantees solution completeness even without this bound. By integrating multi-objective optimization theory with temporal graph modeling, the proposed framework achieves both broad applicability and practical utility.

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.

Problem

• multiobjective
• temporal shortest path
• label correcting
• non-monotonic objectives
• zero-duration cycles

Innovation

• label correcting
• multiobjective temporal shortest path
• non-monotonic objectives
• zero-duration cycles
• path length bound