Independent Set Reconfiguration Under Bounded-Hop Token
Pith reviewed 2026-05-23 22:49 UTC · model grok-4.3
The pith
Independent set reconfiguration under the k-Jump model has identical computational complexity for every k at least 3 on general graphs.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The computational complexity of the ISReconf under the k-Jump model for general graphs is equivalent for all k >= 3. A polynomial-time algorithm solves the ISReconf under the 2-Jump model for split graphs. The optimization variant computing the minimum number of steps is NP-complete for chordal graphs of diameter at most 2k + 1, for any k >=3.
What carries the argument
The k-Jump model, a reconfiguration rule that replaces one vertex in an independent set with another vertex at distance at most k while preserving independence.
If this is right
- Reachability questions under any fixed k >= 3 have the same complexity status as under the unlimited Token Jumping model.
- The polynomial-time algorithm for 2-Jump on split graphs separates the complexity from the 1-Jump case on the same graphs.
- Shortest reconfiguration sequences remain NP-hard to compute even when the input graph has diameter bounded by a constant multiple of k.
Where Pith is reading between the lines
- The hardness threshold appears to lie between distance 2 and distance 3, so bounded-hop variants collapse to the unbounded case once the bound reaches 3.
- The split-graph polynomial algorithm for k=2 might extend to other classes where the 1-Jump version is known to be hard.
- The diameter restriction in the NP-completeness result for optimization suggests similar hardness may hold on graphs with larger but still linear diameter.
Load-bearing premise
Any yes or no instance under a k-Jump model with k larger than 3 reduces to an equivalent instance under the 3-Jump model.
What would settle it
A pair of independent sets on some graph that can be reconfigured under 4-Jump but not under 3-Jump would disprove the claimed equivalence.
read the original abstract
The independent set reconfiguration problem (ISReconf) is the problem of determining, for given independent sets I_s and I_t of a graph G, whether I_s can be transformed into I_t by repeatedly applying a prescribed reconfiguration rule that transforms an independent set to another. As reconfiguration rules for the ISReconf, the Token Sliding (TS) model and the Token Jumping (TJ) model are commonly considered. While the TJ model admits the addition of any vertex (as far as the addition yields an independent set), the TS model admits the addition of only a neighbor of the removed vertex. It is known that the complexity status of the ISReconf differs between the TS and TJ models for some graph classes. In this paper, we analyze how changes in reconfiguration rules affect the computational complexity of reconfiguration problems. To this end, we generalize the TS and TJ models to a unified reconfiguration rule, called the k-Jump model, which admits the addition of a vertex within distance k from the removed vertex. Then, the TS and TJ models are the 1-Jump and D(G)-Jump models, respectively, where D(G) denotes the diameter of a connected graph G. We give the following three results: First, we show that the computational complexity of the ISReconf under the k-Jump model for general graphs is equivalent for all k >= 3. Second, we present a polynomial-time algorithm to solve the ISReconf under the 2-Jump model for split graphs. We note that the ISReconf under the 1-Jump (i.e., TS) model is PSPACE-complete for split graphs, and hence the complexity status of the ISReconf differs between k = 1 and k = 2. Third, we consider the optimization variant of the ISReconf, which computes the minimum number of steps of any transformation between Is and It. We prove that this optimization variant under the k-Jump model is NP-complete for chordal graphs of diameter at most 2k + 1, for any k >=3.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript introduces the k-Jump model for independent-set reconfiguration (ISReconf), which generalizes token sliding (k=1) and token jumping (k=diameter). It claims three results on general graphs: (1) the decision problem has equivalent complexity for every k >= 3; (2) the 2-Jump variant on split graphs is solvable in polynomial time (contrasting with PSPACE-completeness for k=1); (3) the optimization variant (minimum-length sequence) is NP-complete on chordal graphs of diameter at most 2k+1, for any fixed k >= 3.
Significance. If the reductions and algorithm are correct, the work unifies the complexity landscape for all sufficiently large hop distances and isolates a complexity threshold between k=1 and k=2 on split graphs. The optimization hardness result on bounded-diameter chordal graphs is also of interest. No machine-checked proofs or reproducible code are mentioned.
major comments (3)
- [Abstract / §1] Abstract and §1: the central claim that ISReconf complexity is identical for all k >= 3 rests on a reduction from arbitrary k > 3 to the 3-Jump case that must preserve exact reachability in the reconfiguration graph. No construction, gadget, or proof sketch is supplied, so it is impossible to verify that the simulation of long jumps by distance-3 jumps neither creates spurious paths nor destroys existing ones when diam(G) > 3.
- [§4] §4 (presumed location of the split-graph algorithm): the polynomial-time claim for 2-Jump on split graphs is stated without an explicit running-time bound, description of the dynamic-programming state, or comparison to the known PSPACE-completeness proof for 1-Jump; the separation therefore cannot be assessed for tightness.
- [§5] §5 (optimization hardness): the NP-completeness reduction for chordal graphs of diameter <= 2k+1 must ensure that the constructed graph remains chordal and satisfies the diameter bound while encoding the original shortest-reconfiguration instance. No reduction details or verification that the diameter bound is preserved appear in the supplied text.
minor comments (2)
- [§2] Notation: the manuscript uses D(G) for diameter but does not explicitly define the k-Jump rule when the graph is disconnected; a clarifying sentence would help.
- [Abstract] The abstract states three concrete results but supplies no proof sketches, reduction details, or algorithm descriptions, lowering immediate verifiability.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below by pointing to the relevant sections and clarifying our approach. We are happy to revise for improved clarity.
read point-by-point responses
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Referee: [Abstract / §1] Abstract and §1: the central claim that ISReconf complexity is identical for all k >= 3 rests on a reduction from arbitrary k > 3 to the 3-Jump case that must preserve exact reachability in the reconfiguration graph. No construction, gadget, or proof sketch is supplied, so it is impossible to verify that the simulation of long jumps by distance-3 jumps neither creates spurious paths nor destroys existing ones when diam(G) > 3.
Authors: The equivalence for all k ≥ 3 is established by a reduction in Section 3 (Theorem 3.1 and Lemmas 3.2–3.4). The gadget simulates a k-jump via a sequence of 3-jumps on an auxiliary path structure attached to the original vertices; the proof shows that any valid 3-jump sequence in the new graph projects to a valid k-jump sequence in the original (and vice versa) because independent-set constraints and distance bounds prevent both spurious configurations and loss of existing paths. A high-level overview appears in §1, but we will expand it with a one-paragraph sketch of the gadget and the reachability argument. revision: partial
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Referee: [§4] §4 (presumed location of the split-graph algorithm): the polynomial-time claim for 2-Jump on split graphs is stated without an explicit running-time bound, description of the dynamic-programming state, or comparison to the known PSPACE-completeness proof for 1-Jump; the separation therefore cannot be assessed for tightness.
Authors: Section 4 presents a dynamic-programming algorithm that exploits the split-graph partition (clique C and independent set I). The DP table is indexed by subsets of at most two tokens in C and their possible 2-neighbors in I; transitions enumerate valid 2-jumps while maintaining independence. We will add the explicit O(n^5) running-time bound, the formal recurrence for the DP states, and a paragraph contrasting the approach with the PSPACE-completeness reduction for token sliding (k=1) on the same class to emphasize the complexity threshold. revision: yes
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Referee: [§5] §5 (optimization hardness): the NP-completeness reduction for chordal graphs of diameter <= 2k+1 must ensure that the constructed graph remains chordal and satisfies the diameter bound while encoding the original shortest-reconfiguration instance. No reduction details or verification that the diameter bound is preserved appear in the supplied text.
Authors: The NP-completeness proof for the optimization variant appears in Section 5. The reduction attaches a constant number of paths and cliques to each vertex of the source instance so that the resulting graph is chordal (no induced cycles of length ≥4 are created) and its diameter is at most 2k+1 (all new distances are bounded by the hop parameter k). The minimum reconfiguration length in the new instance equals the original optimum plus a fixed additive constant. We will insert a short paragraph after the construction that explicitly verifies chordality and the diameter bound. revision: partial
Circularity Check
No circularity: standard complexity reductions and algorithms with no self-referential reductions or fitted predictions.
full rationale
The paper claims equivalence of ISReconf complexity for k-Jump models when k>=3 via reductions, a P-time algorithm for k=2 on split graphs, and NP-completeness for an optimization variant on bounded-diameter chordal graphs. These are established through explicit algorithmic constructions and polynomial-time reductions between problem instances, which are independent of the target results and do not reduce by construction to fitted parameters, self-citations, or renamed inputs. No equations, ansatzes, or uniqueness theorems are invoked in a self-referential manner. The derivation chain is self-contained against external benchmarks in complexity theory.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Undirected simple graphs with standard notions of distance, independent set, and diameter.
invented entities (1)
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k-Jump model
no independent evidence
discussion (0)
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