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arxiv: 2510.15593 · v2 · submitted 2025-10-17 · 💻 cs.DS

Temporal Graph Reconfiguration for Always-Connected Graphs

Paul Sievers , George Skretas , Georg Tennigkeit This is my paper

Pith reviewed 2026-05-18 06:13 UTC · model grok-4.3

classification 💻 cs.DS
keywords temporal graphsgraph reconfigurationdynamic programmingAPX-hardnessspanning treesalways-connected graphstemporal networksconnectivity preservation
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The pith

Reconfiguring always-connected temporal graphs by single edge time changes can be decided with dynamic programming.

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

The paper introduces reconfiguration problems for temporal graphs where edge activation times must be altered gradually. It focuses on the Layered Connectivity Reconfiguration problem, which asks whether one always-connected temporal graph can be transformed into another by changing the activation time of one temporal edge per step while ensuring every intermediate graph stays always-connected. A dynamic programming algorithm is given to decide if such a sequence exists. The work also establishes that computing the shortest sequence is APX-hard and proves equivalence to the Spanning Tree Sequence Reconfiguration problem, which supplies a simpler algorithm and an inapproximability result for that problem.

Core claim

The Layered Connectivity Reconfiguration problem on always-connected temporal graphs admits a dynamic programming algorithm for deciding the existence of a valid sequence. Finding the shortest reconfiguration sequence is APX-hard. The LCR problem is equivalent to the Spanning Tree Sequence Reconfiguration problem, thereby providing a simpler algorithm for STSR and proving that STSR is APX-hard.

What carries the argument

The dynamic programming algorithm that tracks reachable states of always-connected temporal graphs under single-edge activation time changes, together with the equivalence mapping to spanning tree sequences.

If this is right

  • The existence of a reconfiguration sequence between two always-connected temporal graphs can be decided in polynomial time.
  • No constant-factor approximation algorithm exists for the shortest reconfiguration sequence unless P equals NP.
  • The Spanning Tree Sequence Reconfiguration problem now has a simpler dynamic programming algorithm.
  • The Spanning Tree Sequence Reconfiguration problem is APX-hard.

Where Pith is reading between the lines

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

  • Network maintenance schedules could use the decision procedure to plan gradual link time adjustments without service loss.
  • The equivalence may let hardness and algorithmic techniques transfer between temporal connectivity problems and classical spanning tree reconfiguration.
  • Similar step-by-step change models could be studied for other temporal requirements such as diameter bounds or flow capacities.

Load-bearing premise

The two input temporal graphs are always-connected and every intermediate graph after a single edge activation time change must also remain always-connected.

What would settle it

A small concrete instance with a handful of vertices and time steps where two always-connected temporal graphs are given, the dynamic programming states are computed, and manual checking shows either a missed valid sequence or an invalid one reported as possible.

read the original abstract

Network redesign problems ask for modifications to the edges of a given graph to satisfy certain properties. In temporal graphs, where edges are only active at certain times, we are sometimes only allowed to modify when the edges are going to be active. In practice, we might not even be able to perform all of the necessary modifications at once; changes must be applied step-by-step while the network is still in operation, meaning that the network must continue to satisfy some properties. To initiate a study in this area, we introduce the class of temporal graph reconfiguration problems. As a starting point, we consider the Layered Connectivity Reconfiguration (LCR) problem: Given two always-connected temporal graphs G1 and G2, determine if it is possible to transform G1 into G2 by changing the time at which a single temporal edge is active in each step, such that every intermediate temporal graph is always-connected. We provide a dynamic programming algorithm for the LCR problem. We also show that finding the shortest reconfiguration sequence between two temporal graphs is APX-hard. Additionally, we show that the LCR problem is equivalent to the Spanning Tree Sequence Reconfiguration (STSR) problem introduced by Hanaka et al. Therefore, our results also answer the two open questions presented by the authors: (i) find a simpler algorithm for the STSR problem, (ii) show that the STSR problem is inapproximable up to some factor.

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

2 major / 2 minor

Summary. The paper introduces the Layered Connectivity Reconfiguration (LCR) problem on always-connected temporal graphs: given two such graphs G1 and G2, decide whether G1 can be transformed into G2 by repeatedly changing the activation time of exactly one temporal edge while ensuring every intermediate graph remains always-connected. It presents a dynamic programming algorithm for the decision version of LCR, proves that computing a shortest reconfiguration sequence is APX-hard, and establishes an equivalence between LCR and the Spanning Tree Sequence Reconfiguration (STSR) problem of Hanaka et al., thereby resolving the two open questions posed in that prior work.

Significance. If the equivalence and DP correctness hold, the work supplies a simpler algorithmic solution for STSR together with an APX-hardness result for shortest sequences, directly answering the open questions from Hanaka et al. and advancing the study of reconfiguration problems that must preserve connectivity invariants at every step.

major comments (2)
  1. The equivalence claim (LCR ≡ STSR) is load-bearing for the resolution of the cited open questions. The proof must exhibit an explicit bijection showing that every valid single-edge time change in an LCR sequence corresponds to a valid STSR move (and vice versa) while preserving the always-connected property in every intermediate layer; any mismatch in the mapping of sequences that keep connectivity would invalidate both the equivalence and the transfer of the DP and hardness results.
  2. The dynamic programming algorithm for LCR reachability must explicitly track per-layer connectivity after each single-edge change. If the state does not maintain an invariant that every intermediate temporal graph remains always-connected, the algorithm may accept invalid sequences or miss valid ones, undermining the claimed solution for both LCR and the equivalent STSR problem.
minor comments (2)
  1. The abstract states that the input consists of two always-connected temporal graphs but does not include a concise formal definition or a small illustrative example of the always-connected property; adding one would improve readability.
  2. The hardness result is stated as APX-hardness for the shortest sequence; the reduction should be checked for whether it produces always-connected instances on both sides, as required by the problem definition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying the key aspects of the equivalence and DP that require clear exposition. Below we respond point by point to the major comments, indicating where we will strengthen the presentation.

read point-by-point responses

  1. Referee: The equivalence claim (LCR ≡ STSR) is load-bearing for the resolution of the cited open questions. The proof must exhibit an explicit bijection showing that every valid single-edge time change in an LCR sequence corresponds to a valid STSR move (and vice versa) while preserving the always-connected property in every intermediate layer; any mismatch in the mapping of sequences that keep connectivity would invalidate both the equivalence and the transfer of the DP and hardness results.

    Authors: We agree that the equivalence rests on a precise bijection. Section 4 already defines a mapping that sends each LCR move (a single temporal-edge activation-time change) to the corresponding STSR tree-edge swap while preserving the spanning-tree property in every layer. The forward direction shows that any always-connected LCR sequence yields a valid STSR sequence; the converse direction shows that any STSR sequence can be realized by a sequence of single-edge time changes that never violates always-connectedness. Because the two problems are defined on identical underlying temporal graphs and the connectivity invariant is identical, the mapping is bijective on valid sequences. We will add a short lemma that isolates the invariant preservation argument and an illustrative figure to make the correspondence fully explicit. revision: partial

  2. Referee: The dynamic programming algorithm for LCR reachability must explicitly track per-layer connectivity after each single-edge change. If the state does not maintain an invariant that every intermediate temporal graph remains always-connected, the algorithm may accept invalid sequences or miss valid ones, undermining the claimed solution for both LCR and the equivalent STSR problem.

    Authors: The DP state in Section 3 encodes the current activation-time assignment together with a per-layer spanning-tree certificate. Each transition is permitted only when the certificate remains valid after the single-edge change, thereby enforcing the always-connected invariant on every intermediate temporal graph. The recurrence therefore never generates an invalid sequence. To address the referee’s concern about explicitness, we will revise the state description and the transition predicate to state the invariant check in a single displayed equation and will include a short proof that every accepted path corresponds to a valid LCR sequence. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces the new LCR problem via explicit definition on always-connected temporal graphs, constructs a dynamic programming algorithm for reachability, proves APX-hardness for shortest sequences via standard reduction techniques, and establishes equivalence to the independently introduced STSR problem from Hanaka et al. (different authors). None of these steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the central claims rest on new constructions and external prior work that is not reproduced or assumed within the paper itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper builds on standard definitions from temporal graph theory and reconfiguration problems; no free parameters, invented entities, or ad-hoc axioms beyond domain assumptions are introduced.

axioms (2)
  • domain assumption A temporal graph consists of a static graph with each edge assigned an activation time.
    Standard definition in temporal graphs used to frame the LCR problem.
  • domain assumption A temporal graph is always-connected if the underlying graph is connected at every time instant.
    Core property required for all graphs in the LCR problem statement.

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Reference graph

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