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Better late, then? The hardness of choosing delays to meet passenger demands in temporal graphs

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Selecting delays to help passengers is P when routes are fixed

desk verdict A promising new temporal-graph problem with a thorough hardness study, but the main tractability proof has a wrong LP inequality that invalidates Theorem 10 as printed, and the FPT bound is understated. read the letter →

arxiv 2501.18987 v2 pith:42SL2PZS submitted 2025-01-31 cs.DS cs.CCcs.DM

David C. Kutner , Anouk Sommer This is my paper · ORCID
classification cs.DScs.CCcs.DM MSC 68Q1768Q2505C8590C05
keywords TemporalgraphsDelaymanagementComputationalcomplexityNP-completenessLinearprogrammingFixed-parametertractabilityTrainnetworkspaths
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces DelayBetter, a temporal-graph decision problem: given edge availability times and passengers each with a starting vertex, destination, and latest arrival time, can some edges be delayed so that every passenger still arrives on time? Its central positive claim is that fixing the route each passenger takes makes the problem tractable: (δ-)Path DelayBetter is asserted to lie in P via a linear program, with DelayBetter on trees following as a corollary and a fixed-parameter tractable (fpt) algorithm parameterized by the feedback edge set and the number of passengers. The hardness side shows that without fixed routes the problem is NP-complete even when only two time steps are allowed, on undirected diameter-6 graphs and on acyclic digraphs, and remains NP-complete on planar bounded-degree graphs at lifetime 19. A sympathetic reader would see this as a first temporal-graph complexity treatment of the operations-research delay management problem.

What carries the argument

The linear program with variables $t'_{uv}$ for every edge. Constraints (2) and (3) keep the delaying within bounds; constraint (4), $t'_{uv} \leq t'_{vw} + 1$, is the mechanism meant to enforce strict temporal paths along each prescribed route; constraint (5) enforces arrival deadlines. The integrality of the LP, stated as Claim 11, is the load-bearing step: it lets the paper move from feasibility of the relaxed LP to an actual delaying, and the same LP is then reused as a subroutine in the feedback-edge-set enumeration.

What would settle it

Build a Path DelayBetter instance with a single demand whose prescribed path consists of two edges e1 then e2 sharing a vertex, give both edges the same initial time t, and choose a deadline equal to t. The LP with the printed inequality admits the integral solution t'_e1 = t'_e2 = t, which is LP-feasible but does not yield a strict temporal path; observing this for any such instance settles whether the claimed exact correspondence between LP solutions and valid delayings in Theorem 10 holds.

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Extended reading notes

Core claim

The paper's core discovery is that route information is the complexity switch. Theorem 10 claims that for Path DelayBetter and δ-Path DelayBetter a single linear program decides feasibility and even optimizes total lateness, because an optimal solution can be taken to be integral; the variables are the delayed time labels $t'_{uv}$, with constraints requiring no edge to be moved earlier, bounding the delay per edge in the δ version, capping each arrival edge at its demand's deadline, and (via Equation (4)) prescribing the order of consecutive edges along each demanded path. Corollary 12 lifts tractability to DelayBetter when the footprint is a tree, since each pair of vertices has a unique path. Theorem 13 further lifts this to graphs of bounded feedback edge set by enumerating the possible orderings and edge subsets per passenger, solving a Path DB instance for each combination. For the unrestricted problem, the paper proves NP-completeness with lifetime 2 (Theorems 14 and 15) and on planar bounded-degree graphs with lifetime 19 (Theorem 17).

Load-bearing premise

The proof of Theorem 10 rests on the assumption that the inequality $t'_{uv} \leq t'_{vw} + 1$ enforces strict time increase along each passenger's prescribed path, but that inequality allows consecutive equal labels; if it is not corrected, an integral LP solution can fail to describe a valid delaying.

Editorial extensions

If this is right

  • If Theorem 10 is correct, deciding whether delays can realize fixed-route passenger demands, and even optimizing total lateness, costs polynomial time.
  • Corollary 12 places (δ-)DelayBetter in P for tree-shaped networks, since each pair of vertices has a unique static path.
  • Theorem 13 gives an fpt algorithm for general DelayBetter parameterized by the feedback edge set size plus the number of passengers, with running time $O(\rho! \cdot 2^{\rho} \cdot |D| \cdot \mathrm{poly}(n))$ on directed graphs and $O(\rho! \cdot 3^{\rho} \cdot |D| \cdot \mathrm{poly}(n))$ on undirected graphs.
  • The lifetime-2 hardness results show that even two available time steps do not trivialize the general problem, on undirected diameter-6 graphs and on acyclic digraphs.
  • The planar hardness result shows that δ-DelayBetter remains NP-complete on planar bounded-degree graphs at lifetime 19 for delays up to 10.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • An optimization counterpart of Path DelayBetter that minimizes total lateness is also polynomial, since the LP objective already sums the demand-by-demand lateness margins.
  • The lifetime-2 hardness indicates that short horizons do not automatically make temporal reachability-modification problems easy; route flexibility is the true source of hardness, a lesson that may transfer to other delay-based modification problems.
  • The tractability boundary suggested by the tree corollary is that acyclic footprints are easy, while the fpt result suggests a feedback-edge-set bound as a useful structural parameter; testing whether treewidth or feedback vertex number also suffice would be a natural next step.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper introduces DelayBetter (DB) and two variants, (path-constrained and delay-bounded), in which one may delay time-edges of a temporal graph so that prescribed passenger demands (source, destination, arrival deadline, and optionally a fixed route) are realized by strict temporal paths. The authors claim polynomial-time algorithms for Path DB and δ-Path DB via a linear program, a corollary for trees, an FPT algorithm parameterized by the feedback edge set and the number of passengers, and NP-completeness results for bounded lifetime and planar instances. The hardness reductions are substantial, but the central tractability proof contains a critical error in the LP constraint that is supposed to enforce strict temporal paths, and the stated FPT running time is incorrect.

Significance. If repaired, the paper would describe a natural temporal-graph formalization of delay management with clean tractability results for path-fixed passengers and for trees, complemented by strong NP-hardness results for very small lifetime and for planar graphs. The problem formulation and the planned complexity dichotomy are interesting and would likely be useful to the temporal graphs and delay management communities. However, the main positive results are not established as written because the key LP constraint does not enforce strictness, and the FPT complexity bound is understated; these issues are central to the paper's contribution.

major comments (3)
  1. [Section 3, Theorem 10, Equation (4)] The constraint t'_uv <= t'_vw + 1 does not enforce strict temporal paths. A strict path requires t'_uv + 1 <= t'_vw, i.e., t'_uv <= t'_vw - 1. As printed, the inequality permits consecutive edges to have equal times (or even decreasing times), so an integral LP solution need not correspond to any valid delaying. For example, for a single demand on path u-v-w with initial times 1 and deadline 2, the LP with the printed inequality admits the solution (t'_uv, t'_vw) = (1,1), which satisfies all constraints but gives no strict temporal path, while a feasible delaying (1,2) exists. This invalidates the claimed equivalence in the proof of Theorem 10, and since Corollary 12 and Theorem 13 rely on Theorem 10, the tractability results as stated are unsupported. A corrected inequality would likely restore the approach, but the proof as written is invalid.
  2. [Section 3, Theorem 13] The claimed running time O(ρ!·2^ρ·|D|·poly(n)) for directed graphs (and O(ρ!·3^ρ·|D|·poly(n)) for undirected graphs) is incorrect. For each ordering of the feedback edge set, each demand independently chooses one of 2^ρ (directed) or 3^ρ (undirected) subsets of E', so the number of Path-DB instances generated is ρ!·(2^ρ)^{|D|} (directed) or ρ!·(3^ρ)^{|D|} (undirected). The correct bounds are O(ρ!·2^{ρ|D|}·poly(n)) and O(ρ!·3^{ρ|D|}·poly(n)), respectively. The algorithm is still FPT in ρ + |D|, but the stated complexity in the theorem and proof is wrong and should be corrected.
  3. [Appendix A.4, Claim 11] The integrality proof of the LP is not rigorous as written. The iterative rounding argument asserts that rounding down a non-integral variable and then repeating leads to a contradiction, but it does not carefully handle the possibility that rounding down one variable may violate a constraint involving another non-integral variable, and that the chain of violations may cycle or terminate without a clear contradiction. Since the LP (after correcting Equation (4)) is a system of difference constraints with a network matrix, a total unimodularity argument would provide a clean proof; as written, Claim 11, which is essential for Theorem 10, rests on a sketch with a gap.
minor comments (4)
  1. [Section 2, Lemma 3] In the undirected gadget proof, the demands are defined as (uv_i, v', i+1) for i in [t, t+δ], but later the text refers to 'some demand (uv_i, v', t)'; the intended demand index is i+1, and the notation should be made consistent.
  2. [Section 3, Theorem 13] The theorem statement contains typos: 'reps.' should be 'resp.', and 'for forδ-DB' should be 'for δ-DB'.
  3. [Section 3, Theorem 10] The aside that a modification of Kahn's algorithm 'has been omitted' is unnecessary because the paper already cites polynomial-time LP solvability; either provide the algorithm or delete the sentence.
  4. [Figure 1] The caption refers to 'the two leftmost passengers' although the text describes three passengers starting at u, v, and w; the caption should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: all central derivations are direct reductions or external-hardness imports; Theorem 10 has a correctness gap in constraint (4), but no claim is equivalent to its input by construction.

full rationale

None of the paper's load-bearing derivations reduces to its own inputs. The tractability result (Theorem 10) is a direct LP encoding of the Path DB definition: variables are delayed times, constraints (2), (3), and (5) transcribe the delaying and deadline conditions, and constraint (4) is intended to transcribe strictness of consecutive edges on each demanded path. This is a formulation, not a circular prediction: feasibility of the LP is not assumed from the instance, and the converse direction explicitly checks that any feasible delaying gives an LP solution. The fpt result (Theorem 13) enumerates feedback-edge-set orderings and path subsets, then calls Theorem 10; it does not presuppose the answer. Hardness proofs reduce from external NP-complete problems (Positive NAE 3SAT, CBP-EPE), so they import hardness from outside and do not assume the target problem's hardness. The only self-citation is [21] in Table 1 and related work for MaxMinTaRDiS; it is contextual and no theorem relies on it. The manuscript does contain a serious non-circular gap: as printed, inequality (4) is t'_uv <= t'_vw + 1, which permits equal consecutive edge times, whereas strict temporal paths require t'_uv + 1 <= t'_vw; this makes the proof of Theorem 10 invalid as written, and the omitted Kahn-algorithm proof is also flagged in the text. These are correctness and completeness concerns, not circularity. Therefore the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces a new decision problem but no fitted constants or new entities. Its central claims rest on standard assumptions about temporal graph models, on external NP-completeness results, and on the correctness of the presented LP-based algorithms, the latter of which is compromised by the error in equation (4).

assumptions (4)
  • domain assumption Strict temporal paths and simple temporal graphs as the model
    The paper explicitly restricts to strict temporal paths and simple graphs; results are claimed for this model, so it is an assumption of the problem setting rather than a hidden ad hoc step.
  • domain assumption Positive Not-All-Equal Exactly 3SAT is NP-complete
    Used as the source problem in Theorems 14 and 15, cited from [11]; if this external hardness result were false, the reductions would fail.
  • domain assumption Cubic Bipartite Planar Edge Precoloring Extension is NP-complete
    Used in Theorem 17 and cited from [24]; external hardness assumption.
  • standard math Linear programming is solvable in polynomial time
    Theorem 10 relies on polynomial LP solving [19] and on Claim 11's integrality assertion; the integrality proof is only sketched in the appendix and does not rescue the incorrect inequality.

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Pith. "Pith review of Better late, then? The hardness of choosing delays to meet passenger demands in temporal graphs." pith.science (2026). https://pith.science/paper/42SL2PZS

@misc{pith2026250118987,
  author       = {Pith},
  title        = {Pith review of: Better late, then? The hardness of choosing delays to meet passenger demands in temporal graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/42SL2PZS}},
  note         = {Machine review of arXiv:2501.18987}
}
abstract

In train networks, carefully-chosen delays may be beneficial for certain passengers, who would otherwise miss some connection. Given a simple (directed or undirected) temporal graph and a set of passengers (each specifying a starting vertex, an ending vertex, and a desired arrival time), we ask whether it is possible to delay some of the edges of the temporal graph to realize all the passengers' demands. We call this problem DelayBetter (DB), and study it along with two variants: in $\delta$-DelayBetter, each delay must be of at most $\delta$; in ($\delta$-)Path DB, passengers also fully specify the vertices they should visit on their journey. On the positive side, we give a polynomial-time algorithm for Path DB and $\delta$-Path DB, and obtain as a corollary a polynomial-time algorithm for DB and $\delta$-DB on trees. We also provide an fpt algorithm for both problems parameterized by the size of the graph's Feedback Edge Set together with the number of passengers. On the negative side, we show NP-completeness of ($1$-)DB on bounded-degree temporal graphs even when the lifetime is $2$, and of ($10$-)DB on bounded-degree planar temporal graphs of lifetime $19$. Our results complement previous work studying reachability problems in temporal graphs with delaying operations. This is to our knowledge the first such problem in which the aim is to facilitate travel between specific points (as opposed to facilitating or impeding a broadcast from one or many sources).

Figures

Figures reproduced from arXiv: 2501.18987 by the authors.

Figure 7
Figure 7. A sketch of our reduction from Cubic Bipartite Planar Edge Precoloring Extension to DelayBetter. Only bold time-edges are labeled. For each vertex v ∈ V (G), we create a vertex-gadget consisting of a copy of v and 12 other vertices s v B, sv R, sv G, v1 B, v2 B, v3 B, v1 R, v2 R, u3 R, v1 G, v2 G, v3 G (subscripts represent color; superscript i represents the ith neighbor of v). These vertices are connected differen… view at source ↗

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Exploring Temporal Graphs with Frequent and Regular Edges

    cs.DS 2025-05 reject novelty 6.0 of 10

    A temporal graph in which every edge appears at least once every F timesteps can be explored in F(2n-3) steps, and this is tight up to an additive constant.

Reference graph

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