Restless Temporal Paths: Models & Algorithms

• Restless temporal paths are time-respecting routes in temporal graphs with additional constraints on waiting times at vertices.
• These models are applied to infection chains, packet routing, collision-free multi-agent motion, signaling pathways, and scheduled transportation.
• Research on restless temporal paths focuses on NP-hardness, fixed-parameter tractability, and the development of innovative algorithms for temporal network analysis.

Restless temporal paths are time-respecting paths in temporal graphs under additional temporal constraints on movement or waiting. Across the literature, the term covers several closely related models: paths whose edge-times are strictly increasing, paths whose edge-times are non-decreasing but whose waiting at intermediate vertices is bounded by a parameter such as $\Delta$ or $\beta$, and variants in which vertex-time occupancy itself is part of the feasibility condition. These models are used for infection chains with finite infectious periods, packet routing with bounded buffer time, collision-free multi-agent motion, signaling pathways, and scheduled transportation (Casteigts et al., 2019, Thejaswi et al., 2020, Klobas et al., 2021, Brunelli et al., 22 Jan 2025).

1. Formal models and terminology

A standard discrete-time temporal graph is written as $G=(V,(E_1,\dots,E_T))$, where $E_i$ is the set of edges present at time $i$. In this layer-based model, a temporal walk from $s=v_0$ to $z=v_k$ is a sequence $\big((v_{i-1},v_i,t_i)\big)_{i=1}^k$ with $\{v_{i-1},v_i\}\in E_{t_i}$ and either $t_i\le t_{i+1}$ in the non-strict setting or $\beta$0 in the strict setting; a temporal path additionally requires that the vertices are pairwise distinct (Klobas et al., 2021, Ibiapina et al., 2022). In the terminology of "journeys", the same idea appears as a sequence of time edges $\beta$1 with $\beta$2, and the last label $\beta$3 is the arrival time (Spirakis et al., 2013).

Several papers use explicit waiting bounds. In the point model, a timed arc is $\beta$4, meaning departure from $\beta$5 at time $\beta$6 and arrival at $\beta$7 at time $\beta$8. A temporal path

$\beta$9

is $G=(V,(E_1,\dots,E_T))$0-restless if

$G=(V,(E_1,\dots,E_T))$1

The interval model replaces point availability by interval arcs $G=(V,(E_1,\dots,E_T))$2, where departure can occur at any $G=(V,(E_1,\dots,E_T))$3, and the same waiting bound is imposed on consecutive departures and arrivals (Cauvi et al., 8 Jul 2025). In the discrete undirected model of restless temporal paths, the condition is written directly on edge labels: $G=(V,(E_1,\dots,E_T))$4 This formulation emphasizes bounded waiting at each intermediate vertex (Casteigts et al., 2019).

A further refinement important for disjointness is temporal occupancy. In the non-strict model of temporally disjoint paths and walks, a walk visits an internal vertex $G=(V,(E_1,\dots,E_T))$5 during the interval $G=(V,(E_1,\dots,E_T))$6, the starting vertex during $G=(V,(E_1,\dots,E_T))$7, and the terminal vertex during $G=(V,(E_1,\dots,E_T))$8. Two walks temporally intersect if they occupy the same vertex during overlapping time intervals; they are temporally disjoint if no such vertex-time overlap exists (Klobas et al., 2021). Closely related is the notion of a temporal vertex $G=(V,(E_1,\dots,E_T))$9, used as the basic unit of temporal cuts and of t-vertex-disjointness (Ibiapina et al., 2022).

The literature also extends restless reachability to colored and attributed settings. In vertex-colored temporal graphs, a path may be required to realize a prescribed multiset of vertex colors while still respecting the waiting bound $E_i$0 (Thejaswi et al., 2020). A related, but distinct, generalization is the temporal beer path: edges are time-dependent and designated beer vertices are active only at specified times, so a feasible path must visit at least one beer vertex during one of its active times (D'Ascenzo et al., 11 Jul 2025).

2. Random temporal models and probabilistic structure

One influential sparse random model assigns each static edge exactly one time label, chosen independently and uniformly from $E_i$1. A temporal graph satisfying this UNI-CASE is called a Uniform Random Temporal Graph (U-RTG), and the normalized version sets $E_i$2 when the graph has $E_i$3 vertices (Spirakis et al., 2013). In a uniform random temporal clique $E_i$4, the probability that a fixed simple path of length $E_i$5 becomes temporal is

$E_i$6

and the expected number of temporal paths of length $E_i$7 is

$E_i$8

For the Hamiltonian case $E_i$9 with $i$0,

$i$1

showing that very long one-shot temporal paths are asymptotically rare even in a clique (Spirakis et al., 2013).

The same paper defines temporal distance via foremost journeys. If $i$2 denotes the arrival time of a foremost $i$3-journey under a given labeling, then a capped temporal distance is

$i$4

From this it defines the maximum expected temporal distance

$i$5

and the temporal diameter

$i$6

For a uniform random temporal star,

$i$7

For the normalized uniform random temporal clique, the greedy Extend-Try procedure yields a journey from $i$8 to $i$9 by time

$s=v_0$0

with probability at least

$s=v_0$1

which in turn gives

$s=v_0$2

for the normalized random temporal clique (Spirakis et al., 2013).

A different strict model appears in the random temporal hypercube. Each edge of $s=v_0$3 receives an i.i.d. continuous weight, and a direct path is accessible if its edge weights are strictly increasing. If $s=v_0$4 is the number of accessible direct paths from $s=v_0$5 to $s=v_0$6, then

$s=v_0$7

where $s=v_0$8 and $s=v_0$9 are independent $z=v_k$0 random variables; equivalently, the limit is a mixed Poisson law with random mean $z=v_k$1. The limiting zero-mass is

$z=v_k$2

where $z=v_k$3 is the Gompertz constant (Eide et al., 23 Sep 2025). This places strict, no-waiting temporal accessibility in a critical probabilistic regime distinct from ordinary reachability.

3. Existence, hardness, and parameterized tractability

Restless reachability becomes hard as soon as waiting is bounded. Restless Temporal Path is NP-complete for all finite $z=v_k$4 and $z=v_k$5, even if every edge has exactly one time stamp; in particular, hardness already holds for $z=v_k$6 and lifetime $z=v_k$7. The problem is also W[1]-hard when parameterized by the distance to disjoint paths of the underlying graph, which implies W[1]-hardness for parameters such as feedback vertex number and pathwidth (Casteigts et al., 2019).

Positive results exist for several parameterizations. For Short Restless Temporal Path, there is a randomized algorithm running in time $z=v_k$8 with one-sided error, and a deterministic algorithm running in time $z=v_k$9 (Casteigts et al., 2019). The same work gives fixed-parameter tractability for the vertex cover number and treedepth of the underlying graph, an algorithm running in time $\big((v_{i-1},v_i,t_i)\big)_{i=1}^k$0 for the feedback edge number $\big((v_{i-1},v_i,t_i)\big)_{i=1}^k$1, and an FPT algorithm parameterized by the timed feedback vertex number. Given a timed feedback vertex set of size $\big((v_{i-1},v_i,t_i)\big)_{i=1}^k$2, that algorithm runs in

$\big((v_{i-1},v_i,t_i)\big)_{i=1}^k$3

and the timed feedback vertex set itself can be computed in time $\big((v_{i-1},v_i,t_i)\big)_{i=1}^k$4; a polynomial-time 8-approximation is also available (Casteigts et al., 2019).

Parameterization above lower bounds further refines the path-length approach. If $\big((v_{i-1},v_i,t_i)\big)_{i=1}^k$5 is the minimum length of a temporal $\big((v_{i-1},v_i,t_i)\big)_{i=1}^k$6-$\big((v_{i-1},v_i,t_i)\big)_{i=1}^k$7 path without the restless constraint, then Short Restless Temporal Path can be solved in randomized time

$\big((v_{i-1},v_i,t_i)\big)_{i=1}^k$8

where $\big((v_{i-1},v_i,t_i)\big)_{i=1}^k$9 is the allowed restless path length. This is an above-lower-bound algorithm in the excess parameter $\{v_{i-1},v_i\}\in E_{t_i}$0 (Zschoche, 2022).

A more recent line studies purely temporal width measures. In the point model with uniform delay $\{v_{i-1},v_i\}\in E_{t_i}$1, point-restless-temporal-path is fixed-parameter tractable parameterized by vertex-IM-width $\{v_{i-1},v_i\}\in E_{t_i}$2, with deterministic running time

$\{v_{i-1},v_i\}\in E_{t_i}$3

and in the point model with arbitrary positive delays the running time becomes

$\{v_{i-1},v_i\}\in E_{t_i}$4

By contrast, in the interval model the problem is NP-hard even when the interval vertex-IM-width equals $\{v_{i-1},v_i\}\in E_{t_i}$5 (Cauvi et al., 8 Jul 2025). This separates point-based and interval-based restless path models sharply.

The corresponding cut problem is harder still. Restless Temporal $\{v_{i-1},v_i\}\in E_{t_i}$6-Separation—deleting at most $\{v_{i-1},v_i\}\in E_{t_i}$7 vertices to destroy all $\{v_{i-1},v_i\}\in E_{t_i}$8-restless temporal $\{v_{i-1},v_i\}\in E_{t_i}$9-$t_i\le t_{i+1}$0 paths—is $t_i\le t_{i+1}$1-complete for all $t_i\le t_{i+1}$2, even if every edge has only one time stamp, and it is W[2]-hard when parameterized by the separator size $t_i\le t_{i+1}$3 (Molter, 2021).

4. Disjoint paths, temporal cuts, and the failure of temporal Menger for paths

For multiple routes, the distinction between walks and paths becomes structural. Two temporal walks are temporally disjoint if they never occupy the same vertex at overlapping times (Klobas et al., 2021). In general graphs, Temporally Disjoint Paths is NP-hard even if $t_i\le t_{i+1}$4 and $t_i\le t_{i+1}$5, whereas Temporally Disjoint Walks is W[1]-hard when parameterized by $t_i\le t_{i+1}$6 but lies in XP with running time

$t_i\le t_{i+1}$7

for fixed $t_i\le t_{i+1}$8 (Klobas et al., 2021). On temporal trees, Temporally Disjoint Paths is FPT with running time

$t_i\le t_{i+1}$9

and on a temporal line with only endpoint pairs it is solvable in

$\beta$00

time (Klobas et al., 2021).

A finer structural classification shows large differences between paths and walks. Temporally Disjoint Walks on temporal lines is FPT with respect to $\beta$01, but on temporal stars it is W[1]-hard with respect to $\beta$02. For the path version, the combination $\beta$03 is W[1]-hard, while $\beta$04 plus the feedback edge number of the underlying graph yields an FPT algorithm (Kunz et al., 2023). Small underlying graph structure therefore does not imply uniform tractability.

The most striking separation from static graph theory is the path analogue of Menger’s theorem. Let $\beta$05 be the maximum number of pairwise t-vertex-disjoint temporal $\beta$06-$\beta$07 paths, and let $\beta$08 be the minimum size of a temporal-vertex cut intersecting every temporal $\beta$09-$\beta$10 path. For temporal walks, the walk-based analogue of Menger’s theorem holds. For temporal paths, however,

$\beta$11

and this is the only case in which equality holds. For every fixed $\beta$12, there exists a temporal graph with

$\beta$13

Algorithmically, deciding whether $\beta$14 is polynomial-time solvable for $\beta$15 and NP-complete for $\beta$16. The cut problem is co-NP-hard and XP in the cut size (Ibiapina et al., 2022). A common static-graph intuition—that max disjoint paths and min separating sets remain dual in temporal settings—therefore fails for paths.

5. Reachability algorithms, centrality, and optimization primitives

Beyond decision problems, restless temporal paths support algorithmic primitives for large-scale reachability. An algebraic framework based on constrained multilinear sieving solves restless reachability and colored restless reachability when parameterized by path length $\beta$17. The proposed problems can be solved in

$\beta$18

time and

$\beta$19

space, where $\beta$20 is the number of vertices, $\beta$21 the number of edges, and $\beta$22 the maximum resting time. The same framework handles vertex-colored temporal graphs with multiset color queries, and an open-source implementation scales to graphs with up to one billion temporal edges (Thejaswi et al., 2020).

Centrality under waiting constraints has also become algorithmically tractable. In the model $\beta$23, a restless temporal walk $\beta$24 satisfies

$\beta$25

between consecutive edges. For temporal betweenness computation on restless walks, an edge-centric framework computes exact betweenness for several optimality criteria—including shortest, foremost, fastest, shortest foremost, and shortest fastest—in $\beta$26 time and $\beta$27 space. Earlier restless computation was known only for the shortest criterion, with complexity $\beta$28. The same work reports exact computations on large temporal graphs with over a million temporal edges and a public-transit case study on six networks, including Berlin, Rome, and Paris (Brunelli et al., 22 Jan 2025).

Optimization on minimal temporal systems also appears. In a two-vertex temporal multigraph interpreted as $\beta$29 parallel "bridges" or, equivalently, one edge with labels $\beta$30, a greedy $\beta$31 algorithm minimizes the maximum load-plus-time cost, and the optimum is

$\beta$32

This is a stylized scheduling problem on a temporally labeled edge and shows that restless temporal routing naturally interfaces with congestion-aware load balancing (Spirakis et al., 2013).

6. Related generalizations, applications, and open directions

A closely related time-window formulation is the temporal beer path problem. Here, temporal edges are quadruples $\beta$33, beer vertices $\beta$34 are active only at times $\beta$35, and a valid path must visit at least one beer vertex while it is active. Four optimization criteria are defined: earliest-arrival, latest-departure, fastest, and shortest temporal beer paths. Efficient algorithms are given under both edge-stream and adjacency-list representations, together with preprocessing and static-graph transformations for dynamic conditions such as new openings or closings of shops (D'Ascenzo et al., 11 Jul 2025). This suggests a broader class of restless-like path problems in which feasibility depends jointly on edge schedules, vertex-time windows, and path optimality.

Several recurring misconceptions are contradicted by current results. Small or simple underlying graphs do not guarantee easy computation: temporal lines and stars already support NP-hardness or W[1]-hardness for disjoint routing (Klobas et al., 2021, Kunz et al., 2023). Bounded temporal width is not uniformly sufficient: vertex-IM-width yields FPT algorithms in the point model but does not rescue the interval model, where hardness persists at width $\beta$36 (Cauvi et al., 8 Jul 2025). And the static intuition behind Menger’s theorem transfers from temporal walks to temporal paths only in the case $\beta$37 (Ibiapina et al., 2022).

Open directions are explicit in several papers. For sparse random temporal graphs, proposed extensions include different randomization models $\beta$38-CASE, multi-labeled U-RTGs, upper bounds for general U-RTGs, and exact values of $\beta$39 and $\beta$40 even for the normalized clique (Spirakis et al., 2013). For temporally disjoint routing, the walks problem on temporal trees remains open, and further structural parameters such as cutwidth or bandwidth are natural next targets (Klobas et al., 2021, Kunz et al., 2023). For the temporal hypercube, convergence of higher moments beyond the first two is left open (Eide et al., 23 Sep 2025). For separation, it is conjectured that Restless Temporal $\beta$41-Separation parameterized by the separator size belongs to the parameterized class $\beta$42 (Molter, 2021).

Taken together, these results show that restless temporal paths are not a single problem but a family of tightly related models at the intersection of temporal reachability, bounded waiting, path simplicity, and vertex-time occupancy. The resulting theory combines exact counting in random temporal structures, sharp hardness frontiers, nontrivial fixed-parameter algorithms, and application-driven generalizations in routing, diffusion, and temporal network analysis.