Label Correcting Algorithms for the Multiobjective Temporal Shortest Path Problem

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.

• The paper introduces two label correcting frameworks, one for isotonic objectives and one for general objectives, detailing distinct pruning and extension strategies.
• It shows that efficient temporal paths may require exponential-length routes with non-simple cycles, especially when monotonicity and isotonicity assumptions fail.
• The research rigorously proves algorithm correctness under bounded and unbounded path length scenarios, offering practical insights for time-critical network applications.

Label Correcting Algorithms for the Multiobjective Temporal Shortest Path Problem

Problem Overview and Formulation

This paper addresses the single-source multiobjective temporal shortest path problem (SSMTSPP) in discrete-time directed temporal graphs. The scenario considers a source node $s$, multiple objectives ($p \geq 1$), and the task of identifying, for every destination node $v$, the set of nondominated images (Pareto optimal solutions) of temporal $s$-$v$ paths along with corresponding efficient paths. Temporal graphs model dynamic connectivity, relevant for time-critical networks such as transportation, where arcs are annotated with time and traversal duration.

The fundamental challenge addressed is generalizing shortest path computations when monotonicity and isotonicity in the objectives cannot be assumed. Traditionally, monotonicity (generalizing nonnegativity of arc costs) and isotonicity (order preservation of objective values upon extension) are pivotal for label setting algorithms. However, many realistic objectives—e.g., price in airline networks affected by hidden-city ticketing—break these assumptions, necessitating more general algorithmic approaches.

Algorithmic Contributions

Label Correcting Frameworks

Two main algorithmic frameworks are analyzed:

1. Isotonic Objectives Algorithm: For isotonic but possibly non-monotone objectives, a label correcting algorithm is constructed. Due to isotonicity, dominated labels can be safely discarded at each iteration, adhering to standard practices in multiobjective static shortest path computation.
2. General Objectives Algorithm: For fully general objectives (not necessarily monotone or isotonic), dominated labels cannot be discarded until finalization, since extensions of dominated labels may lead to nondominated images. Labels are extended and retained throughout the iterative process, and only after all paths up to length $K$ are considered, dominated labels are pruned.

Both algorithms use a parameter $K$ that restricts admissible path length, motivated by examples demonstrating that efficient solutions may require paths of arbitrarily large length, especially with zero-duration cycles (temporal cycles traversable in zero total time), which can be exploited to attain certain Pareto solutions.

Algorithmic Correctness

Strong correctness guarantees are provided:

• For the bounded-length variant (SSMTSPP-MPL), both frameworks provably enumerate all $K$-efficient nondominated images for each destination node.
• For isotonic objectives, dominated labels can be discarded iteratively; for non-isotonic cases, only at the algorithm's end.
• Under additional sufficient conditions (e.g., absence of zero-duration cycles reachable from source, positive minimum waiting times), one can guarantee completeness and correctness for all efficient solutions using $|R|$ (number of arcs) iterations.

Structural Insights and Complexity

A significant technical result is that, unlike classical additive-cost static graphs, in temporal graphs with general objectives, the shortest and most efficient paths may need to traverse cycles an arbitrarily large (even exponential in graph size) number of times. Examples illustrate that efficient solutions can require paths of exponential length and even non-simple cycles (constructed from multiple overlapping simple cycles).

The existence of improving cycles (cycles that, when traversed, continually enhance some objective) fundamentally affects problem tractability; their presence can signal infinitely many nondominated images or even the necessity for unbounded path length. The paper demonstrates that certifying the absence of improving cycles reduces the admissible path length, regaining tractability reminiscent of static multiobjective shortest path frameworks.

Numerical and Theoretical Results

• In scenarios with finite sets of attainable objective values at each node, the total number of required iterations is polynomially bounded.
• For rational additive objectives, a variant of the main algorithm identifies improving cycles and exhaustively computes Pareto solutions within $m$ iterations.
• Algorithms are proven correct by induction, leveraging multiobjective dominance principles. Stopping criteria (no changes in label sets upon iteration) assure completeness without excess computation.

Implications and Future Directions

The algorithmic schemes extend multiobjective temporal shortest path computation beyond previous works that impose monotonicity and isotonicity (cf. "A general label setting algorithm and tractability analysis for the multiobjective temporal shortest path problem" [Bazgan+etal:monotone]). The paper shows that temporal graph problems with general objectives require fundamentally different treatment, both practically (accounting for real-world phenomena where monotonicity fails) and theoretically (addressing exponential path length and non-simple cycle requirements).

Practically, the findings suggest that optimization in dynamic time-indexed networks (e.g., logistics, transportation) can be computationally intensive when objective structures are only weakly constrained. Theoretical implications include the necessity to redefine tractability boundaries for temporal multiobjective optimization, highlighting gaps between static and temporal settings.

Potential future work includes:

• Complexity analysis for improving cycle detection in temporal graphs with general objectives.
• Identification of additional sufficient conditions for algorithmic completeness without bounded $p \geq 1$0.
• Algorithmic improvements for the all-pairs multiobjective temporal shortest path problem, which arises in many temporal network applications.

Conclusion

This paper develops general-purpose multiobjective label correcting algorithms that operate effectively in temporal graphs with arbitrary objective structures, relaxing key assumptions traditionally used in shortest path computation. The results bridge the gap between theory and practical application, demonstrating the impact of objective properties on tractability and algorithm design, and laying a foundation for robust optimization in highly dynamic temporal network environments.