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REVIEW 2 major objections 5 minor 23 references

Fully Packed and Ready to Go: High-Density, Rearrangement-Free, Grid-Based Storage and Retrieval

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Rearrangement-free storage is guaranteed for any grid whose open side is at least three columns wide.

desk verdict Sharp c≥3 characterization with a real proof gap in the only-if direction; worth reviewing but needs fixes. read the letter →

arxiv 2505.22497 v1 pith:Y4YDA467 submitted 2025-05-28 cs.RO cs.DS

Tzvika Geft , Kostas Bekris , Jingjin Yu This is my paper
classification cs.ROcs.DS
keywords grid-basedstoragerearrangement-freehigh-densitypuzzle-basedonlinelookaheadrelocationautomatedwarehousingcolumn-adjacentpaths
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 asks whether a rectangular warehouse grid can be filled to capacity and later emptied without ever moving a load out of the way. Its central answer is a sharp boundary: if the open side of the grid is at least three cells wide, a rearrangement-free plan exists for every arrival and departure order, and the paper gives a linear-time placement algorithm. If the open side is only one or two cells wide, some instances inevitably require relocations. The same ideas extend to online settings: a lookahead of $3r-1$ arrivals suffices for full-density no-relocation packing, and with no lookahead at all, density must be capped at $2/3$ to guarantee one-action access. A sympathetic reading is that the classic density-versus-rearrangement tradeoff largely disappears once a modest amount of open access is available.

What carries the argument

The argument reduces each storage instance to two departure sequences that one layout must satisfy: the true departure order and the reverse of the arrival order. Observation 3 then turns this into a local condition: every load not on the front row must be adjacent to a load that departs earlier. Algorithm 1 exploits this by assigning the next unmatched loads of the two sequences to the first two columns and matched loads to a third column, and the proof verifies the adjacency conditions column by column. The online and approximation results additionally rely on column-adjacent access paths, meaning each storage or retrieval path lies mostly in one column with at most one short lateral segment, which makes the plans efficient and robot friendly.

What would settle it

A systematic search over all arrival permutations for a $3 \times 2$ or $4 \times 2$ grid would settle the question: if even one instance at full density admits a relocation-free plan, the exact boundary in Theorem 2 is wrong.

Watch

Extended reading notes

Core claim

The central claim is Theorem 2: for an $r \times c$ storage area with $r > 1$, a solution that avoids relocations exists for every arrival and departure sequence if and only if $c \geq 3$. The constructive direction is proved by Algorithm 1, which builds the layout bottom-up while maintaining local adjacency conditions, and by a column-by-column reduction that fills all but the last three columns with sorted batches of loads. The negative direction rests on an obstruction for $2 \times 2$ grids that the paper asserts extends to all $r \times 2$ grids. If the theorem holds, any grid with a three-cell opening can be operated at 100 percent density with exactly one storage and one retrieval action per load, with no reshuffling.

Load-bearing premise

The 'only if' half of the exact $c \geq 3$ threshold depends on the assertion that every grid with exactly two columns inherits the obstruction shown for a $2 \times 2$ grid, and that extension is stated rather than proved.

Editorial extensions

If this is right

  • Any $r \times c$ warehouse with $c \geq 3$ can be run at 100 percent density with exactly $2n$ actions, one storage and one retrieval per load, regardless of the arrival order.
  • The $O(n)$ placement algorithm makes the guarantee practical to compute at warehouse scale.
  • With a lookahead of $3r-1$ arrivals, the offline guarantee survives: the grid can still be filled to capacity without relocations, using column-adjacent paths.
  • With a lookahead of 1 and at most $r(c-1)+1$ loads, one column can be kept nearly empty and all moves remain relocation-free, giving an optimal strategy for sparse storage.
  • In the fully online setting, guaranteeing at most one action per load forces density down to at most $2/3$, matching the familiar aisle-based warehouse layout.

Reading between the lines

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

  • The 3-column threshold suggests a design rule of thumb for new storage grids: if the arrival or departure schedule is known in advance, opening the grid on at least three cells buys full-density operation with no relocation cost, so the extra width is effectively free.
  • Theorem 3's column-filling strategy is naturally parallelizable for the first $c-3$ columns, so multi-robot execution could reach near-peak throughput; the paper mentions this but does not formally analyze congestion or scheduling.
  • The local-adjacency condition may transfer to other one-sided access geometries such as shelves, where loads that depart earlier could be placed adjacent to later-departing loads to avoid occlusion moves; this is an extension, not a claim of the paper.
  • A testable practical extrapolation is that automated parking garages with three-cell-wide entry bays could adopt fully packed layouts without shuffle moves, though interleaved arrivals and departures are outside the paper's strict two-phase model.
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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

2 major / 5 minor

Summary. The paper studies a grid-based storage system in which unit loads occupy cells of an r×c grid, the bottom row is the only access side, and a robot can move loads through empty cells along cardinal directions. The problem StoRe2 asks whether all loads can be stored in a prescribed arrival order and later retrieved in a prescribed departure order using exactly one action per load, i.e., without relocations. The authors characterize the offline case: for r>1, a rearrangement-free solution exists if and only if c≥3, and they give an O(n)-time placement algorithm for c≥3. They then consider online variants: with lookahead 3r−1, full-capacity no-relocation storage is possible for c≥3; with lookahead 1, r(c−1)+1 loads can be stored without relocations, and for r≤c a 1.125-approximation for the minimum-action problem is claimed. For the fully online setting they derive a density/action tradeoff of 2a/(2a+1) for a worst-case bound of a actions per load. The paper closes with experiments on square grids comparing the proposed algorithm with a best-first baseline.

Significance. If the characterization holds, the paper gives a strong positive result: for any arrival and departure sequences, 100% density is achievable with exactly two actions per load once the access side is three cells wide, contrasting with stack-based systems where related problems are NP-hard. The constructive algorithms are concrete, run in linear or near-linear time, and produce column-adjacent paths that are attractive for practical robot deployment. The paper also carefully separates information regimes (offline, lookahead, fully online) and connects the fully-online tradeoff to Gue's density bound. The main reservations are the unproved necessity direction of the central characterization and an omitted proof of the 1.125-approximation; both are fixable but currently prevent the claims from being fully verified.

major comments (2)
  1. [Section IV, Observation 1 and Theorem 2] The 'only if' direction of Theorem 2 is not established. The text supports it only by Observation 1 for a 2×2 grid and the sentence 'The observation extends to any r×2 case where r≥3.' The 2×2 forcing argument does not carry over to taller grids: with additional rows, the second column need not behave as a single stack, and for r=3 the very sequence A=(1,4,2,3), D=(1,2,3,4) used in Observation 1 admits a rearrangement-free arrangement (for example, place 1, 2, 4 in the left column bottom-to-top and 3 in the front-right cell). Thus the extension requires a separate proof, and Theorem 2's iff characterization is not proven as written. Please supply a rigorous proof for all r≥3, or correct the statement if it is false.
  2. [Section V, Theorem 4] The 1.125-approximation claim is stated with 'The proof is omitted.' Since this is a listed contribution, the omission is load-bearing. Moreover, the preceding accounting (one action per storage, at most two actions per retrieval, at most r−1 relocations) does not transparently imply a 9/8 bound; one must count the total number of actions relative to the 2n lower bound. Please include the full calculation, for example by showing that the total number of actions is at most 2rc+r−1 and that this is within 9/8 of the optimum when r≤c, and clarify whether the r−1 relocations are included in or additional to the 'at most two actions per retrieval' count.
minor comments (5)
  1. [Section III, action definitions] In the definition of Relocation, 'The empty call v' should read 'The empty cell v.'
  2. [Section III/IV, notation] The problem name is introduced as StoRe2 but later appears as StoRE2; please standardize the capitalization.
  3. [Section IV, Observation 3] Observation 3 states an 'if and only if' condition for an arrangement satisfying a departure sequence, but the necessity argument assumes full density (no initially empty cells). Please state this assumption explicitly, since at partial densities an initially empty cell can provide access without an earlier-departing neighbor.
  4. [Figure 4 caption] The notation D′=(12_1, 7_2, ...) with subscripts is not defined; please explain what the subscripts indicate.
  5. [Section VII, Table I] The experiments average over 25 instances but report no variance. Please include standard deviations or confidence intervals, and state the exact lower bound used for the action suboptimality metric.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the constructive offline proofs and fully-online constructions are self-contained; the unproved r×2 only-if extension is a correctness gap, not a self-referential step.

full rationale

The paper's central claims are derived constructively rather than from their own conclusions. Theorem 1 gives an explicit Algorithm 1 that assigns loads to C1, C2, and C3 and then verifies the local adjacency conditions of Observation 3 by the order of placement; Observation 2 is a reversible-path equivalence, not an assumption of the result. Theorem 3 extends the construction to c≥3 by filling extra columns in departure order and then invoking Theorem 1 on the last three columns, so the if-direction of Theorem 2 is self-contained. The only-if direction rests on Observation 1 for the 2×2 case and on the unproved sentence 'The observation extends to any r×2 case where r≥3'; this is a genuine proof gap and a correctness risk, but it is not circularity, because the assertion neither assumes the target theorem nor is fitted to the data that it is meant to explain. In the fully online part, the paper imports Gue's upper bound ρ(k)≤2k/(2k+1) as an external theorem [12] and supplies its own matching aisle-based constructions (Propositions 2 and 3), so the tradeoff is not renamed into existence. The authors' earlier works ([10], [15]) appear only as related-work or practical-setting citations and are not load-bearing. No equation or parameter is defined in terms of the prediction it supports, and no fitted input is relabeled as a prediction. Therefore the derivation chain is free of circularity, even though the r×2 only-if step needs an independent proof.

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

The paper introduces no new objects or fitted numbers. Its load-bearing inputs are the model definition (whole-bottom-row access, cardinal moves through empty cells), the local adjacency characterization (Observation 3), the w.l.o.g. relabeling D=[n], and Gue's external upper bound for the fully-online density tradeoff. The only risk in this ledger is the unverified applicability of Gue's bound to the relaxed access model.

assumptions (4)
  • domain assumption An arrangement satisfies a departure sequence if and only if every load is either in the bottom row or adjacent to a load that departs earlier (Observation 3).
    This is the key geometric characterization on which Algorithm 1's correctness rests; it depends on the free-motion model where a load can move through any empty cell.
  • standard math Loads can be relabeled so the departure sequence is [n] without loss of generality.
    Since loads are distinguishable, renaming them does not change the problem; this lets the paper fix D = (1,...,n).
  • domain assumption Gue's upper bound rho(k) <= 2k/(2k+1) applies to the relaxed setting with whole-bottom-row access (cited as Theorem 5).
    Used for Proposition 3; if the bound is specific to single-I/O connected free-space layouts, the fully-online density/action tradeoff needs re-examination.
  • domain assumption The grid is accessible via the entire bottom row; a robot can move loads along any path of empty cells in four cardinal directions.
    This is the problem definition from Section III; all algorithm correctness and the c>=3 characterization rely on this motion capability.

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Cite this review

Pith. "Pith review of Fully Packed and Ready to Go: High-Density, Rearrangement-Free, Grid-Based Storage and Retrieval." pith.science (2026). https://pith.science/paper/Y4YDA467

@misc{pith2026250522497,
  author       = {Pith},
  title        = {Pith review of: Fully Packed and Ready to Go: High-Density, Rearrangement-Free, Grid-Based Storage and Retrieval},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y4YDA467}},
  note         = {Machine review of arXiv:2505.22497}
}
read the original abstract

Grid-based storage systems with uniformly shaped loads (e.g., containers, pallets, totes) are commonplace in logistics, industrial, and transportation domains. A key performance metric for such systems is the maximization of space utilization, which requires some loads to be placed behind or below others, preventing direct access to them. Consequently, dense storage settings bring up the challenge of determining how to place loads while minimizing costly rearrangement efforts necessary during retrieval. This paper considers the setting involving an inbound phase, during which loads arrive, followed by an outbound phase, during which loads depart. The setting is prevalent in distribution centers, automated parking garages, and container ports. In both phases, minimizing the number of rearrangement actions results in more optimal (e.g., fast, energy-efficient, etc.) operations. In contrast to previous work focusing on stack-based systems, this effort examines the case where loads can be freely moved along the grid, e.g., by a mobile robot, expanding the range of possible motions. We establish that for a range of scenarios, such as having limited prior knowledge of the loads' arrival sequences or grids with a narrow opening, a (best possible) rearrangement-free solution always exists, including when the loads fill the grid to its capacity. In particular, when the sequences are fully known, we establish an intriguing characterization showing that rearrangement can always be avoided if and only if the open side of the grid (used to access the storage) is at least 3 cells wide. We further discuss useful practical implications of our solutions.

Figures

Figures reproduced from arXiv: 2505.22497 by the authors.

Figure 1
Figure 1. Application examples. Left: Grid-based storage using robotic vehicles (AGVs) for transferring loads [5]. The AGV can go beneath a load and can move in all four grid directions. Right: Illustration of an automated parking garage where vehicles are the load to be autonomously placed and retrieved [7]. Similar to the first case, AGVs can go under vehicles to transport them. components (thereby slowing down delivery) or… view at source ↗
Figure 2
Figure 2. A solution without relocations for an input arrival sequence A = (9, 4, 7, 3, 6, 2, 1, 8, 5) and departure sequence D = (1, 2, 3, 4, 5, 6, 7, 8, 9) for a (3 × 3) grid accessible only from the bottom. Snapshots are illustrated from left to right. (a) The first arriving (load) 9 can be directly stored at the top using a (straight) upward path empty. (b) Next, 4 arrives and can be stored in front of 9, leaving the spac… view at source ↗
Figure 3
Figure 3. Consider an instance with A = (1, 4, 2, 3) and D = (1, 2, 3, 4). Given that load 1 must depart first, it has to be stored in the front to avoid relocations. This forces the above-shown storage sequence (or its vertical mirror, where 1 is placed on the bottom right). This leaves load 2 buried behind loads 3 and 4. This means it cannot be retrieved without a rearrangement. Observation 1. For a 2 × 2 storage space relo… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Running Algorithm 1 on the inputs D ′ = (12 1 , 7 2 , 3 3 , 1 4 , 10 5 , 8 6 , 9 7 , 11 8 , 6 9 , 4 10 , 2 11 , 5 12 ) (top row) and D ′ = (8 1 , 3 2 , 4 3 , 5 4 , 6 5 , 2 6 , 7 7 , 12 8 , 1 9 , 10 10 , 9 11 , 11 12 ) (bottom row). Top: (a)(b) Partial solutions after t…
Figure 5
Figure 5. Figure 5: An example execution of the algorithm described in Theorem 3, which also works for proving Theorem 2. A = (4, 10, 6, 12, 2, 3, 9, 15, 1, 14, 13, 7, 5, 11, 8) and D = [15]. (a) Knowing the first three loads to arrive are 4, 10, 6, we store them in the order they depart,…
Figure 6
Figure 6. Figure 6: The storage strategy of Theorem 4 for a (5 × 8) grid. Paths π1–π3 are the left three vertical columns. Paths π4–π7 are L-shaped and distinguished using different colors. π8 passes only the bottom right cell. Corner cells are marked with crosses, and the possible types …
Figure 7
Figure 7. Figure 7: Left: A 4 × 6 storage area with 1-deep aisles. Right: A 4 × 10, 2-deep aisle arrangement. Proof. Consider an aisle-based configuration where each grid column is either completely full of loads or empty. For each set of 2k + 1 contiguous columns, starting from the leftm…

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