Multiobjective Temporal Shortest Path Problem

• The paper establishes new formulations for evaluating temporal paths with multiple objectives, integrating Pareto efficiency with time feasibility in directed discrete-time graphs.
• It introduces methodologies such as MO-SIPP and tropical Dijkstra that efficiently compute nondominated path images while managing dynamic obstacles and varying arc costs.
• Empirical results demonstrate significant improvements, including up to 63% reduction in journey times and enhanced load utilization in congested, real-world networks.

Searching arXiv for the cited papers and closely related formulations. The multiobjective temporal shortest path problem denotes a family of path problems on temporal or time-augmented graphs in which feasible paths must respect time and are evaluated under several objectives simultaneously. In the single-source formulation, given a directed discrete-time temporal graph $G=(V,R)$, a start node $s\in V$, and $p\geq 1$ objectives, the task is, for each $v\in V$, to compute the set of nondominated images of temporal $s$-$v$ paths together with one corresponding efficient path for each image; alternatively, one may have to detect the existence of an improving zero-duration cycle (Marica et al., 7 May 2026). Closely related formulations model dynamic obstacles by safe intervals (Ren et al., 2021), encode timetable uncertainty through interval arc costs and scenario sets (Masing et al., 2024), or treat concurrent routing queries as mutually interacting through edge loads in temporal load-aware road networks (Conlan et al., 2021).

1. Core definitions and solution concepts

A directed discrete-time temporal graph is defined as $G=(V,R)$, where $V$ is a finite set of nodes and $$R\subseteq V\times V\times \mathbb{Q}_{\ge 0}\times \mathbb{Q}_{\ge 0}$$ is a finite set of arcs. Each arc $r$ has a start node $s\in V$0, an end node $s\in V$1, a start time $s\in V$2, and a traversal time $s\in V$3. A temporal path is a sequence

$s\in V$4

such that $s\in V$5, $s\in V$6, and $s\in V$7 for each $s\in V$8; a zero-arcs path $s\in V$9 is also allowed (Marica et al., 7 May 2026).

The objective model in the general single-source setting is also explicit. Each objective is a tuple

$p\geq 1$0

where $p\geq 1$1 is a totally ordered set, $p\geq 1$2 assigns arc-values, $p\geq 1$3 is associative with left-neutral $p\geq 1$4, and $p\geq 1$5 specifies preference direction. For $p\geq 1$6 objectives, the image of a path $p\geq 1$7 is

$p\geq 1$8

Weak domination $p\geq 1$9 holds componentwise according to each objective direction, and strict domination $v\in V$0 means weak domination with inequality in at least one component. An $v\in V$1-$v\in V$2 path is efficient if no other $v\in V$3-$v\in V$4 path strictly dominates its image; the image of an efficient path is nondominated (Marica et al., 7 May 2026).

In the safe-interval formulation for dynamic obstacles, the same Pareto logic is expressed on a time-augmented graph $v\in V$5. For a single agent with start $v\in V$6 and goal $v\in V$7, the aim is to find all collision-free paths $v\in V$8 that minimize $v\in V$9 nonnegative criteria simultaneously, with cost vector

$s$0

A path $s$1 is Pareto-optimal if there is no other collision-free $s$2 with $s$3, and the target set is

$s$4

This formulation makes explicit that temporal feasibility and Pareto efficiency can be imposed simultaneously even when occupancy constraints vary over time (Ren et al., 2021).

2. Temporal modeling variants

The literature represented here uses several temporal abstractions, each tailored to a different source of temporal heterogeneity.

Formulation	Temporal representation	Additional state
SSMTSPP	Directed discrete-time temporal graph $s$5	Arc start times and traversal times
MO-SIPP / MO-CBS	Time-augmented graph $s$6 with safe intervals	Vertex occupancies and interval states
Collective routing	Static directed graph $s$7 with continuous time and discrete partition $s$8	Edge–Load–Matrix $s$9
Tropical Dijkstra	Directed event-activity graph	Interval arc costs $v$0 and scenarios

In the load-aware road-network model, the topology remains static while costs become time- and load-dependent. The graph is $v$1, time is continuous $v$2 with a discrete partition $v$3, and each edge $v$4 carries a capacity $v$5, free-flow travel-time $v$6, speed-limit $v$7, and length $v$8. The edge-load function $v$9 records the number of vehicles scheduled to traverse edge $G=(V,R)$0 in interval $G=(V,R)$1, and these values are collected in the Edge–Load–Matrix (ELM), $G=(V,R)$2 (Conlan et al., 2021).

In the safe-interval model, time is discrete and occupancy is externalized as dynamic obstacles. For each vertex $G=(V,R)$3, the occupied times are $G=(V,R)$4, and safe intervals are the maximal contiguous intervals in $G=(V,R)$5. A SIPP state is therefore not simply a vertex-time pair but a pair $G=(V,R)$6, where $G=(V,R)$7 is one of the non-overlapping safe intervals at $G=(V,R)$8 (Ren et al., 2021).

In the interval-cost passenger-routing model, the graph consists of events and activities. Nodes represent departure or arrival events of a vehicle at a stop, while arcs represent a driving leg, a dwelling time, a transfer between lines, or an access/egress between a street location and a stop. Each arc $G=(V,R)$9 has an interval cost $V$0, and a scenario is a choice $V$1 for every arc. This shifts the temporal dimension from explicit time expansion toward a scenario-parametric shortest-path model (Masing et al., 2024).

A plausible implication is that the term “multiobjective temporal shortest path problem” encompasses not a single canonical encoding but a class of models in which temporal feasibility, uncertain travel times, dynamic occupancy, or endogenous congestion determine how efficient paths are defined and computed.

3. Dominance structure, objective interactions, and complexity

A central distinction in the recent theory is between monotonicity and isotonicity. An objective is monotone when extending a path cannot improve its value in the preferred direction merely by appending an arc; it is isotonic when the order of two path values is preserved if both paths are extended by the same arc. Under both properties, Dijkstra-style label-setting is possible. Without them, zero-duration temporal cycles may generate infinitely many strictly better images or may have to be traversed an arbitrarily large finite number of times to obtain certain nondominated images (Marica et al., 7 May 2026).

This observation rules out a common static-graph intuition. When monotonicity or isotonicity is missing, one cannot fix in advance a small upper bound, such as $V$2, on the length of all efficient paths. The restricted problem variant SSMTSPP-MPL therefore imposes a maximum admissible path length $V$3 and considers only paths containing at most $V$4 arcs. The same paper establishes several sufficient conditions under which such a bound is not required: if no zero-duration cycle reachable from $V$5 exists, then $V$6 suffices; if each node enforces a positive minimum waiting time $V$7, then again $V$8 suffices; if at each node there are at most $V$9 distinct images of $$R\subseteq V\times V\times \mathbb{Q}_{\ge 0}\times \mathbb{Q}_{\ge 0}$$0-$$R\subseteq V\times V\times \mathbb{Q}_{\ge 0}\times \mathbb{Q}_{\ge 0}$$1 paths, both algorithms stabilize within $$R\subseteq V\times V\times \mathbb{Q}_{\ge 0}\times \mathbb{Q}_{\ge 0}$$2 iterations (Marica et al., 7 May 2026).

The load-aware collective formulation introduces a different objective interaction. For $$R\subseteq V\times V\times \mathbb{Q}_{\ge 0}\times \mathbb{Q}_{\ge 0}$$3 queries $$R\subseteq V\times V\times \mathbb{Q}_{\ge 0}\times \mathbb{Q}_{\ge 0}$$4, one seeks paths $$R\subseteq V\times V\times \mathbb{Q}_{\ge 0}\times \mathbb{Q}_{\ge 0}$$5 that collectively optimize three desiderata: minimize total or average travel-time,

$$R\subseteq V\times V\times \mathbb{Q}_{\ge 0}\times \mathbb{Q}_{\ge 0}$$6

ensure fairness through the congestion penalty

$$R\subseteq V\times V\times \mathbb{Q}_{\ge 0}\times \mathbb{Q}_{\ge 0}$$7

and balance edge loads through

$$R\subseteq V\times V\times \mathbb{Q}_{\ge 0}\times \mathbb{Q}_{\ge 0}$$8

A weighted-sum or lexicographic multiobjective can be formed, although the reported implementation focuses on minimizing $$R\subseteq V\times V\times \mathbb{Q}_{\ge 0}\times \mathbb{Q}_{\ge 0}$$9 while reporting fairness and load-utilization as secondary metrics (Conlan et al., 2021).

The same collective routing problem is stated to be NP-hard. The argument given is that even in the static case, routing many origin-destination demands so as to minimize total travel-time under capacity constraints is known to be NP-hard, and the addition of time and dynamic congestion only enriches the problem (Conlan et al., 2021). In the interval-cost setting, complexity is instead governed by the size of the label set $r$0, yielding a worst-case running time

$r$1

for tropical Dijkstra (Masing et al., 2024).

4. Algorithmic paradigms

When all objectives are monotone and isotonic and the first objective is earliest-arrival, a label-setting algorithm can be used. Labels are prioritized by the first objective, permanent labels are stored per node, and temporally feasible outgoing arcs are relaxed only if the resulting label is not dominated. The complexity bound reported for this setting is

$r$2

where $r$3 is the total number of permanent labels and $r$4 is the average out-degree (Marica et al., 7 May 2026).

If isotonicity holds but monotonicity does not, the problem is handled by a multiobjective temporal label-correcting algorithm for SSMTSPP-MPL. It maintains, for each node $r$5 and each length $r$6, a set $r$7 of candidate labels with path-length exactly $r$8, and dominated labels may still be discarded immediately because isotonicity guarantees that they cannot lead to new nondominated images. If neither monotonicity nor isotonicity holds, a fully general label-correcting algorithm keeps all generated labels of length at most $r$9 and removes dominated labels only once no label-sets grow any further; its worst-case running time is

$s\in V$00

with space $s\in V$01 (Marica et al., 7 May 2026).

A different low-level strategy appears in the dynamic-obstacle setting. MO-SIPP associates each state $s\in V$02 with a set of labels $s\in V$03, where $s\in V$04 is the cost-to-come vector and $s\in V$05 is the arrival time. The algorithm uses vector dominance for goal pruning and a specific label-dominance relation at the same state,

$s\in V$06

component-wise. The stated theorem is that, upon termination, MO-SIPP generates exactly the set $s\in V$07 of all cost-unique Pareto-optimal paths from $s\in V$08 to $s\in V$09 in the presence of dynamic obstacles (Ren et al., 2021).

The interval-cost formulation replaces vector labels by tropical polynomials. For each node $s\in V$10, tropical Dijkstra maintains a label set $s\in V$11 of non-dominated polynomials. The path monomial of a path $s\in V$12 is

$s\in V$13

and the path polynomial of a path set $s\in V$14 is

$s\in V$15

Dominance is defined pointwise over all scenarios, and the algorithm returns complete or essential sets by changing the dominance relation used for pruning (Masing et al., 2024).

The load-aware road-network variant extends Dijkstra and A* by replacing static edge weights with a FIFO-compliant arrival-time function

$s\in V$16

where $s\in V$17 and

$s\in V$18

The paper states that this gives a FIFO network, implies that subpaths of shortest paths are shortest, and ensures that no waiting at a node can improve arrival time; the worst-case time for the Temporal–Load–Aware A* variant is $s\in V$19 (Conlan et al., 2021).

5. Specialized formulations: safe intervals, interval scenarios, and collective reassignment

The safe-interval line of work is motivated by collision avoidance in dynamic environments. In that setting, waiting is explicitly represented as a graph action, each edge has a fixed $s\in V$20-vector cost, waiting for one time-step incurs $s\in V$21, and the objective is to enumerate all cost-unique Pareto-optimal collision-free paths for a single agent. MO-SIPP is then embedded in MO-CBS as the low-level search procedure for multi-objective conflict-based search, yielding a formulation referred to as multi-objective MAPF (MOMAPF) (Ren et al., 2021).

The interval-cost line of work starts from public transportation. The Complete Interval Shortest Path Problem asks, for each target $s\in V$22, for the union of all $s\in V$23 shortest paths over all scenarios,

$s\in V$24

whereas the Essential Interval Shortest Path Problem seeks a scenario-covering minimal subset $s\in V$25 such that for every scenario at least one path in $s\in V$26 is shortest. The paper further states that C-ISPP and E-ISPP can be cast as a multiobjective shortest-path problem with $s\in V$27-dimensional costs, in which C-ISPP corresponds to all weakly-efficient paths and E-ISPP to a smallest efficient cover of the Pareto front (Masing et al., 2024).

The collective routing line of work changes the problem semantics more substantially. A $s\in V$28-th query $s\in V$29 departs $s\in V$30 at time $s\in V$31 and must reach $s\in V$32 via a path $s\in V$33. Because assigned paths update the ELM and therefore future travel times, shortest-path queries are no longer isolated tasks. The CS-MAT heuristic repeatedly selects, from a batch of unassigned queries, the query whose earliest-feasible load-aware shortest path gives the lowest arrival time at its destination; once that path is assigned, the ELM is updated and only queries whose previously computed best path intersected the newly assigned path in space-time are re-evaluated. The method is described as “almost embarrassingly parallel” and is reported to scale to hundreds of thousands of queries on modest clusters (Conlan et al., 2021).

This suggests a useful conceptual distinction. Some formulations seek the full nondominated image set for a fixed source or source-goal pair, while others optimize a system-wide objective over many temporally interacting paths. Both remain temporal and multiobjective, but the latter couples path computations through shared network state.

6. Empirical findings, misconceptions, and open questions

The empirical record in these papers is heterogeneous because the tasks differ. In the collective load-aware setting, experiments on Porto and New York datasets during the AM peak report that Average Journey Time was reduced by up to $s\in V$34 over naïve free-flow Dijkstra and by up to $s\in V$35 over static load-aware (SLAD); absolute savings are reported as approximately $s\in V$36 minutes per trip in Porto and approximately $s\in V$37 minutes in New York. The same study reports lower mean and lower standard deviation of the congestion penalty $s\in V$38 than TLAA* under high congestion, increases of up to $s\in V$39–$s\in V$40 in Free-Flow Capacity Utilization and Load Distribution relative to baselines, and routine use of more than $s\in V$41 of network capacity under heavy load. For $s\in V$42 queries in NYC with $s\in V$43 cores, end-to-end runtime including journey is reported as $s\in V$44 minutes for CS-MAT, $s\in V$45 minutes for TLAA*, and $s\in V$46 minutes for Dijkstra (Conlan et al., 2021).

In the safe-interval setting, MO-SIPP was compared, for single-agent performance, to a standard multi-objective A* (NAMOA*) over $s\in V$47. On benchmark grids with random risk fields and $s\in V$48 objectives, it achieved an order-of-magnitude reduction in planning time per call and similar or smaller frontier sizes because of time-dimension compression via safe intervals. The paper states, for example, a reduction from $s\in V$49 seconds to $s\in V$50 seconds on a room map with $s\in V$51 (Ren et al., 2021).

In the tropical Dijkstra study, computational experiments were carried out on $s\in V$52 condensed event-activity instances derived from the Wuppertal public-transport system and $s\in V$53 TimPassLib instances. Small networks with $s\in V$54 finished in milliseconds even without transfer caps; the largest Wuppertal instance, with approximately $s\in V$55 arcs, took approximately $s\in V$56 minutes per source for the complete set and approximately $s\in V$57 minutes for the essential set. Restricting to at most $s\in V$58 transfers sped up all instances by one to two orders of magnitude, making full city-scale networks solvable in less than $s\in V$59 minute per source. The number of complete paths per OD pair ranged from fewer than $s\in V$60 to approximately $s\in V$61, essential sets were $s\in V$62–$s\in V$63 of complete sets on Wuppertal, and empirically $s\in V$64 stayed in the low tens (Masing et al., 2024).

Two misconceptions are directly contradicted by the available results. First, a static-graph bound on efficient path length does not carry over automatically to temporal multiobjective routing; Example 1 in the label-correcting paper shows that an efficient path can require exponential length in $s\in V$65 when objectives are non-monotone, and Example 2 shows that non-simple combinations of zero-duration cycles may be needed to generate a new nondominated image (Marica et al., 7 May 2026). Second, dominated labels cannot always be discarded immediately: this is valid for isotonic objectives, but when isotonicity is absent the fully general label-correcting algorithm keeps all generated labels until stabilization before deleting dominated ones (Marica et al., 7 May 2026).

Open questions are also explicit. For tropical Dijkstra, the stated problems include tight worst-case bounds on $s\in V$66 for structured networks, integration into combined timetabling-routing optimizers such as preprocessing in modulo network-simplex timetabling, and handling truly dynamic costs or time-dependent intervals (Masing et al., 2024). In the broader temporal multiobjective setting, the role of zero-duration cycles, the need for admissible bounds $s\in V$67, and the distinction between monotone, isotone, and fully general objectives remain the principal structural fault lines (Marica et al., 7 May 2026).