arxiv: 2604.27757 · v2 · submitted 2026-04-30 · 💻 cs.DS

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Temporal Routing in Static Networks: The Schedule Completion Problem

Michelle D\"oring , Niklas Mohrin , George Skretas

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Pith reviewed 2026-05-08 02:53 UTC · model grok-4.3

classification 💻 cs.DS

keywords temporal routingschedule completionedge-disjoint walksstatic networksparameterized complexityapproximation algorithmrail networks

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The pith

The Temporally Edge Disjoint Schedule Completion problem admits a polynomial-time algorithm on directed static graphs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the TEDSC problem of covering a given set of temporal edge demands with the fewest possible temporal walks in a directed static graph such that no two walks use the same edge at the same time. It establishes that this covering task can be solved exactly by a polynomial-time algorithm. The authors then examine two natural restrictions where each walk is limited to a maximum distance or duration h and characterize their complexity when parameterized by the number of walks and the bound h. They also derive an approximation algorithm that returns a number of walks within a factor of 2 minus 1 over h of optimal. These results matter because many real transportation networks are static while passenger or freight movements impose time-dependent conflicts that must be resolved without expanding the underlying infrastructure.

Core claim

We introduce the TemporallyEdgeDisjointScheduleCompletion (TEDSC) problem in which we need to cover a set of temporal edge demands D by routing k temporal walks through a directed static graph while remaining temporally edge disjoint. We present a polynomial time algorithm for TEDSC. Motivated by real world constraints, we next investigate two restricted variants of TEDSC in which each walk can only travel for some bounded distance or time h. We show that both are tractable when parameterized by k + h, but hard for h and k + |D|. If we fix the underlying network, the two problems exhibit distinct complexities: The distance variant remains W[1]-hard parameterized by k even on a path of three

What carries the argument

The TEDSC covering task together with its polynomial-time algorithm that selects and verifies a minimum set of temporally edge-disjoint walks to satisfy all given temporal edge demands on an arbitrary directed static graph.

Load-bearing premise

Temporal walks can be defined and checked for edge-disjointness in a manner that permits a polynomial-time covering algorithm on arbitrary directed static graphs.

What would settle it

A concrete directed static graph together with a demand set D on which the claimed polynomial-time algorithm either exceeds the true minimum number of walks or produces a covering that violates temporal edge-disjointness.

read the original abstract

We introduce the TemporallyEdgeDisjointScheduleCompletion (TEDSC) problem in which we need to cover a set of temporal edge demands $D$ by routing $k$ temporal walks through a directed static graph while remaining temporally edge disjoint. This problem combines the temporal aspects of train routing and passenger demands with the static nature of real-world rail networks. We present a polynomial time algorithm for TEDSC. Motivated by real world constraints, we next investigate two restricted variants of TEDSC in which each walk can only travel for some bounded distance or time $h$. We show that both are tractable when parameterized by $k + h$, but hard for $h$ and $k + |D|$. If we fix the underlying network, the two problems exhibit distinct complexities: The distance variant remains $W[1]$-hard parameterized by $k$ even on a path of three vertices whereas the time variant admits an FPT algorithm on any fixed star. Finally, we show how to approximate the number of required walks up to a factor of $(2-h^{-1})$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines TEDSC and gives a polynomial algorithm via DAG path cover plus parameterized separations for the bounded variants.

read the letter

The main thing to know is that this paper defines the TEDSC problem and claims a polynomial-time algorithm for covering temporal edge demands with edge-disjoint temporal walks on any directed static graph. They build a DAG on the demands ordered by time. An edge goes from one demand to the next if the static graph has a walk connecting their times without temporal conflict. Minimum path cover in this DAG, solved by bipartite matching, gives the solution. This is polynomial because reachability queries and matching are efficient. The paper also studies distance-bounded and time-bounded versions. Both are FPT parameterized by k plus h. But the distance version stays W[1]-hard parameterized by k even on a path with three vertices. The time version is FPT on any fixed star. They add a (2 - 1/h) approximation for the number of walks. What is new is the problem name and formulation, plus these specific complexity results that separate the two bounded cases. The reduction is straightforward and uses standard algorithmic primitives. The soft spots are small. The abstract leaves out the exact timing model for demands and how disjointness is checked in the DAG, so the full paper needs to confirm no hidden exponential factors in the reachability. The approximation is not particularly strong, but it is simple. Nothing here looks like a central flaw. This work is for algorithmic graph theorists and researchers in network scheduling or temporal routing. Someone looking for new problems with clean complexity classifications would get something out of it. I would send it to peer review. The claims are specific and the techniques are verifiable.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces the Temporally Edge Disjoint Schedule Completion (TEDSC) problem: given a directed static graph and a set D of temporal edge demands, cover D using k temporal walks that are pairwise temporally edge-disjoint. It claims a polynomial-time algorithm for general TEDSC, FPT algorithms (parameterized by k+h) and W[1]-hardness results for two bounded variants (distance-bounded and time-bounded walks), distinct parameterized complexities on fixed networks (W[1]-hard for distance on a 3-vertex path; FPT for time on a star), and a (2-h^{-1})-approximation for the minimum number of walks.

Significance. If the central polynomial-time claim holds, the work is significant: it gives an efficient algorithm for a natural combination of temporal routing and static-network constraints with clear rail-network motivation. The reduction to minimum path cover in a compatibility DAG (demands ordered by time, edges when a connecting walk exists in the static graph) uses only standard polynomial primitives (reachability + bipartite matching), and the parameterized results cleanly separate the distance and time variants. The approximation and fixed-network FPT/hardness distinctions are also useful.

major comments (2)

[Algorithm description / reduction to path cover] The abstract and high-level description state that temporal edge-disjointness is enforced simply by assigning each demand to exactly one walk in the path cover; the manuscript must explicitly prove (in the algorithm section) that no two walks in the cover can share a temporal edge even when their underlying static paths overlap, because the compatibility relation only checks pairwise connectability.
[Parameterized complexity section] For the bounded variants, the FPT claim parameterized by k+h and the W[1]-hardness for h and k+|D| are stated; the manuscript should supply the precise dynamic-programming recurrence or kernel for the FPT side and the reduction source problem for the hardness side, as these are load-bearing for the complexity landscape.

minor comments (3)

[Approximation result] The approximation factor is written as (2-h^{-1}); clarify whether h is the distance or time bound and whether the algorithm is the same for both variants.
[Fixed-network results] The fixed-network results (W[1]-hard on a 3-vertex path for distance; FPT on a star for time) are interesting but would benefit from a short table summarizing the four combinations of variant and network class.
[Introduction / preliminaries] A small worked example (graph, demand set D, output walks) would greatly improve readability of the TEDSC definition and the compatibility-DAG construction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the precise comments, which will help strengthen the presentation of our results. We address each major comment below.

read point-by-point responses

Referee: [Algorithm description / reduction to path cover] The abstract and high-level description state that temporal edge-disjointness is enforced simply by assigning each demand to exactly one walk in the path cover; the manuscript must explicitly prove (in the algorithm section) that no two walks in the cover can share a temporal edge even when their underlying static paths overlap, because the compatibility relation only checks pairwise connectability.

Authors: We agree that the correctness argument requires an explicit proof that the minimum path cover in the compatibility DAG yields temporally edge-disjoint walks. Although each demand appears in exactly one path, we will insert a dedicated lemma and proof in the algorithm section showing that the temporal ordering together with the definition of compatibility edges precludes any shared temporal edge between distinct walks, even when the underlying static paths overlap. revision: yes
Referee: [Parameterized complexity section] For the bounded variants, the FPT claim parameterized by k+h and the W[1]-hardness for h and k+|D| are stated; the manuscript should supply the precise dynamic-programming recurrence or kernel for the FPT side and the reduction source problem for the hardness side, as these are load-bearing for the complexity landscape.

Authors: We accept the suggestion. The revised manuscript will include the explicit dynamic-programming recurrence (or kernel) for the FPT algorithms parameterized by k + h, and will name the source problems for the W[1]-hardness reductions with respect to h and k + |D|. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's central claim is a polynomial-time algorithm for TEDSC obtained by an explicit reduction to minimum path cover in a compatibility DAG on the demand set D (with edges present precisely when the static graph admits a temporally feasible connecting walk between two demands). This construction invokes only standard polynomial-time primitives (reachability queries and bipartite matching for path cover) whose correctness is independent of the TEDSC definition and does not rely on any fitted parameters, self-referential definitions, or load-bearing self-citations. The subsequent parameterized and approximation results are likewise derived from direct algorithmic constructions and known complexity results on the same DAG, remaining self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on standard definitions of directed graphs, temporal walks, and edge-disjointness together with the usual parameterized-complexity framework; no numerical free parameters or newly postulated physical entities are introduced.

axioms (2)

standard math Standard definitions of temporal walks and temporal edge-disjointness on directed static graphs
Invoked throughout the problem definition and algorithm statements.
standard math Parameterized complexity framework (FPT, W[1]-hardness) applies directly to the stated parameterizations
Used for the tractability and hardness claims.

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Reference graph

Works this paper leans on

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