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arxiv: 2605.11873 · v1 · submitted 2026-05-12 · 💻 cs.DS

Recognition: 2 theorem links

· Lean Theorem

Maximizing Reachability via Shifting of Temporal Paths

Argyrios Deligkas , Michelle D\"oring , Eduard Eiben , George Skretas , Georg Tennigkeit
Authors on Pith no claims yet

Pith reviewed 2026-05-13 04:55 UTC · model grok-4.3

classification 💻 cs.DS
keywords temporal graphsreachability maximizationtemporal pathsshifting operationsparameterized complexityFPT algorithmstrain networksXP algorithms
0
0 comments X

The pith

Reachability maximization in temporal graphs of k paths is fixed-parameter tractable by k alone even with unbounded shifts.

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

The paper studies how to increase the number of nodes reachable from a source in a temporal graph by shifting the time labels on its k underlying paths. Each path must keep strictly increasing times after the shifts, which models adjusting schedules in networks like trains while preserving connectivity along each route. It establishes fixed-parameter tractability when the parameter is k, whether or not the total shift budget b is limited, and also when both k and b are parameters. In nearly all other parameterization regimes the problem is hard yet solvable in XP time. This matters for networks whose structure consists of a modest number of time-dependent routes, where limited schedule adjustments can improve overall accessibility.

Core claim

When a temporal graph is the union of k temporal paths, reachability from a given source can be maximized by shifting the paths' labels while preserving strict temporal order on each path; the resulting problem is fixed-parameter tractable when parameterized by k (even with b unbounded) and by the pair (k, b), while other natural parameterizations admit matching XP algorithms and intractability lower bounds.

What carries the argument

Shifting operations applied uniformly to the labels of each of the k temporal paths that preserve strict temporal continuity along every path, used to optimize source reachability under a budget b on total shifts performed.

If this is right

  • An FPT algorithm exists when both k and b are parameters.
  • An FPT algorithm exists when the parameter is only k, even if b is left unbounded.
  • The problem admits an XP algorithm but is intractable when parameterized by b alone.
  • The problem admits an XP algorithm but is intractable when parameterized by k together with a bound on b.

Where Pith is reading between the lines

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

  • In transportation networks whose routes form only a few paths, modest schedule adjustments can be computed efficiently to improve connectivity from key stations.
  • The tractability for unbounded b indicates that the path-union structure itself supplies enough restriction for efficient search over possible collective shifts.
  • The same shifting model could be tested on maximizing other temporal objectives such as earliest arrival times or flow between multiple source-sink pairs.

Load-bearing premise

The input graph must be exactly the union of k temporal paths whose labels can be shifted while keeping strict increasing order on each path individually.

What would settle it

A W[1]-hardness proof for the problem parameterized solely by k with b unbounded, or a polynomial-time algorithm for parameterization by b alone.

read the original abstract

We examine the problem of maximizing the reachability of a given source in temporal graphs that are given as the union of k temporal paths, i.e., every given path is a sequence of edges with strictly increasing labels that denote availability in time. This type of temporal graphs represent train networks. We consider shifting operations on the labels of the paths that maintain their temporal continuity. This means that we can move the availability of a temporal edge later or earlier in time, and propagate the shifts to all other affected edges of the path in order to preserve its temporal connectivity. We study the parameterized complexity of the problem with respect to the number of paths k, and the total budget b, where b is the maximum number of shifts we are allowed to perform. Our results reveal that fixed parameter tractability can be achieved (1) when parameterized both by k and b, and (2) when parameterized by k, and b is unconstrained. In almost every other case, e.g., parameterized by a single parameter or parameterized by k, while having a bound on b, we establish intractability lower bounds that are matched by XP algorithms.

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 manuscript studies the parameterized complexity of maximizing reachability from a designated source in a temporal graph that is the union of exactly k temporal paths. Shifts are applied to the integer time labels of entire paths (propagating to preserve strict temporal order on each path) subject to a total budget b on the number of shifts performed. The central results are FPT membership when parameterized by the pair {k, b} and when parameterized by k alone (b unbounded), together with matching W[1]-hardness or para-NP-hardness lower bounds (and XP algorithms) for all other natural parameterizations, including single-parameter cases and the regime of bounded b with parameter k.

Significance. If the claimed dichotomy holds, the work supplies a clean and tight parameterized-complexity classification for a natural reachability-maximization problem on temporal graphs that model train networks. The FPT result for parameterization by k with unbounded b is noteworthy, as it permits arbitrary shifts while restricting only the number of paths; the matching upper and lower bounds across regimes strengthen the contribution. The paper also receives credit for stating the model explicitly (union of k temporal paths, strict temporal order after shifts, reachability evaluated post-shift) and for aligning the tractability boundaries with standard techniques such as configuration DP over the k paths.

minor comments (3)
  1. [Abstract] Abstract: the phrase 'in almost every other case' is imprecise; an explicit enumeration of the studied parameterizations (e.g., k alone with b bounded, b alone, etc.) would improve readability without lengthening the abstract.
  2. [Introduction] The manuscript would benefit from a short table or paragraph in the introduction that lists all parameterizations considered together with the corresponding complexity status (FPT / W[1]-hard / para-NP-hard / XP).
  3. [Model / Preliminaries] Notation: the distinction between the original temporal labels and the shifted labels should be introduced with a consistent symbol (e.g., t_e vs. t'_e) at the first use in the model section to avoid later ambiguity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. The referee's summary correctly captures the problem setting, the FPT results for parameterization by {k, b} and by k (with b unbounded), and the matching hardness results for other regimes. We are pleased that the contribution is viewed as supplying a clean dichotomy for this reachability-maximization problem on temporal graphs.

Circularity Check

0 steps flagged

No significant circularity; standard parameterized complexity classification

full rationale

The paper establishes FPT membership via explicit algorithms (configuration DP over the k paths when parameterized by k+b or by k alone) and hardness via standard reductions from known problems, with XP algorithms matching the lower bounds. These steps rely on the explicitly stated model (union of k temporal paths, integer shifts preserving strict temporal order, reachability after shifts) without any self-definitional reduction, fitted-parameter renaming, or load-bearing self-citation chain. The derivation chain is self-contained against external benchmarks and does not reduce any claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the modeling assumption that every input is a union of k temporal paths and that shifts preserve strict temporal order; no free parameters, invented entities, or non-standard axioms are introduced.

axioms (2)
  • domain assumption Input is the union of k temporal paths with strictly increasing time labels
    Explicitly stated in the abstract as the graph class under study.
  • domain assumption Shifts maintain temporal continuity of each path
    Required for the operation to be valid; stated in the abstract.

pith-pipeline@v0.9.0 · 5506 in / 1393 out tokens · 37406 ms · 2026-05-13T04:55:06.661700+00:00 · methodology

discussion (0)

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Reference graph

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