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arxiv: 2607.00118 · v2 · pith:5EASIT5Bnew · submitted 2026-06-30 · 💻 cs.DS

Temporal Path Covers: Dilworth Properties and Parameterized Complexity

Lapo Cioni , Sotiris Kanellopoulos , Edouard Nemery , Aris Pagourtzis , Christos Pergaminelis , Manolis Vasilakis This is my paper

Pith reviewed 2026-07-03 21:59 UTC · model grok-4.3

classification 💻 cs.DS
keywords temporal path coverDilworth propertyLovász numberparameterized complexitytemporal graphsW[1]-hardnessNP-hardness
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The pith

Under the temporal Dilworth property, the minimum temporal path cover size equals the Lovász number of the connectivity graph.

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

The paper studies the Minimum Temporal Path Cover and Minimum Temporally Disjoint Path Cover problems on temporal graphs. It establishes that when the input is promised to satisfy the temporal Dilworth property, the minimum cover size equals exactly the Lovász number of the connectivity graph. This equality yields polynomial-time algorithms for both problems. The authors further prove parameterized hardness and tractability results for various structural parameters.

Core claim

Under the promise that the temporal Dilworth property holds, the size of the minimum T(D)PC is exactly equal to the Lovász number of the connectivity graph, making both problems polynomial-time solvable. TPC is W[1]-hard parameterized by deletion distance to linear forest even for temporal graphs with two time-steps, while an FPT algorithm exists when using vertex cover number together with the number of time-steps; both problems remain NP-hard for constant vertex cover size without the number of time-steps.

What carries the argument

The temporal Dilworth property, which equates minimum T(D)PC size to maximum antichain size, and its exact equivalence to the Lovász number of the connectivity graph under that promise.

If this is right

  • Both TPC and TDPC are polynomial-time solvable under the temporal Dilworth promise.
  • TPC is W[1]-hard parameterized by deletion distance to linear forest even with two time-steps.
  • FPT algorithms exist when parameterizing by vertex cover number plus number of time-steps.
  • Both problems are NP-hard even for constant vertex cover size when the number of time-steps is omitted from the parameter.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The Lovász number computation becomes the effective bottleneck for solving the path cover problems when the promise holds.
  • The W[1]-hardness for deletion distance to linear forest shows that treewidth-based XP algorithms cannot be improved to FPT without stronger parameters.
  • The necessity of including the number of time-steps for tractability with vertex cover suggests similar requirements may apply to other temporal problems.

Load-bearing premise

The input temporal graph satisfies the temporal Dilworth property by promise.

What would settle it

A temporal graph that satisfies the temporal Dilworth property yet has minimum T(D)PC size different from the Lovász number of its connectivity graph.

Figures

Figures reproduced from arXiv: 2607.00118 by Aris Pagourtzis, Christos Pergaminelis, Edouard Nemery, Lapo Cioni, Manolis Vasilakis, Sotiris Kanellopoulos.

Figure 2
Figure 2. Figure 2: Setting K := 2kB + n completes the construction of the instance of TPC. ▶ Lemma 17. If the Restricted Unary Bin Packing instance is positive, then G has a temporal path cover of size K. Proof. Consider a feasible assignment of the items. For every item i, write f(i) = {c, d} with c < d [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
read the original abstract

The Minimum Temporal Path Cover (TPC) and Minimum Temporally Disjoint Path Cover (TDPC) problems were introduced by [Chakraborty, Dailly, Foucaud, Klasing, MFCS '24]. Both were shown to be NP-hard on temporal DAGs, while the latter is also NP-hard on temporal oriented trees. All tractable cases for T(D)PC established in that paper satisfy a temporal Dilworth property, namely that the size of the minimum T(D)PC is equal to the size of the maximum antichain. This raises a natural question: is T(D)PC polynomial-time solvable under the promise that the respective Dilworth property holds? In this work, we answer this question in the affirmative for both problems, proving in fact that, under the respective promise, the size of the minimum T(D)PC is exactly equal to the Lov\'asz number of the connectivity graph. In another direction, we establish parameterized algorithms and hardness results for TPC and TDPC. Our main result is that TPC is W[1]-hard parameterized by the deletion distance to linear forest even for temporal graphs with two time-steps, answering in the negative an open question by Chakraborty et al. about whether an XP algorithm parameterized by treewidth plus number of time-steps can be improved to FPT. On the other hand, we prove that an FPT algorithm does exist if the vertex cover number is used as parameter instead of the treewidth in the above parameterization. We complement this with a proof that including the number of time-steps in the parameter is necessary to yield tractability, as, otherwise, both TPC and TDPC remain NP-hard even for constant vertex cover size. Along the way, we establish various other para-NP-hardness results involving structural parameters such as the pathwidth and the maximum degree of the underlying graph.

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.

Referee Report

0 major / 3 minor

Summary. The paper studies Minimum Temporal Path Cover (TPC) and Minimum Temporally Disjoint Path Cover (TDPC) on temporal graphs. Under the explicit promise that a temporal graph satisfies the temporal Dilworth property (minimum cover size equals maximum antichain size), it proves that this common value equals the Lovász number of the connectivity graph, yielding a polynomial-time algorithm via semidefinite programming. It further shows that TPC is W[1]-hard parameterized by deletion distance to a linear forest (even for two time-steps), that an FPT algorithm exists when parameterizing by vertex cover number together with the number of time-steps, that both problems remain NP-hard for constant vertex cover when the number of time-steps is not in the parameter, and additional para-NP-hardness results for parameters such as pathwidth and maximum degree.

Significance. If the central equality holds, the work affirmatively resolves the open question posed by Chakraborty et al. on tractability under the temporal Dilworth promise and supplies an explicit polynomial-time algorithm via the Lovász number. The parameterized results answer another open question in the negative and delineate precise boundaries between FPT and hard cases for structural parameters on temporal graphs. The explicit reduction to an SDP whose optimum matches both the cover size and the antichain size is a clean contribution that links temporal path covers to a well-studied graph parameter.

minor comments (3)
  1. [Abstract] Abstract: the phrase 'the connectivity graph' is used without a one-sentence definition or pointer to its formal introduction; adding this would improve readability for readers outside the immediate sub-area.
  2. [Introduction / main theorem statement] The statement that the Lovász number is 'polynomial-time computable' should cite the standard reference (e.g., Grötschel et al.) and briefly note that the SDP can be solved to sufficient precision for the integer optimum under the Dilworth promise.
  3. [Parameterized hardness section] The W[1]-hardness reduction for deletion distance to linear forest (two time-steps) is described at high level; a short paragraph clarifying how the temporal edges are placed to enforce the linear-forest deletion set would help verification.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. The report correctly captures the main contributions, including the resolution of the open question on tractability under the temporal Dilworth promise via the Lovász number and the parameterized complexity results. Since no specific major comments were raised, we have no points requiring rebuttal or clarification at this stage.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The central claim establishes a new equality (min T(D)PC size = Lovász number of connectivity graph) under an explicit external promise (temporal Dilworth property) taken from independent prior work by Chakraborty et al. The argument proceeds via a direct SDP reduction whose optimum is shown to match both quantities; this is not a renaming, self-definition, or fitted-input prediction. Parameterized results are separate hardness/FPT proofs with no load-bearing self-citations. No step reduces by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a theoretical computer science paper on graph algorithms and complexity; it introduces no free parameters, ad hoc axioms, or invented entities beyond standard definitions from graph theory and parameterized complexity.

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