Pith. sign in

REVIEW 1 major objections 1 minor 50 references

Dominance-Based Feasibility Inference for Packing-Constrained Pickup and Delivery Problems

T0 review · 1 major / 1 minor · reviewed 2026-06-25 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read An order-preserving mapping infers packing feasibility for pickup-delivery routes from verified reference plans without full rechecks.

desk verdict The order-preserving mapping for dominance inference is the real novelty but its safety under non-monotonic packing needs an explicit argument. read the letter →

arxiv 2606.25970 v1 pith:36ULCR2U submitted 2026-06-24 math.OC

Siqiao Li , Mahnam Saeednia , Patrick Stokkink This is my paper · ORCID
classification math.OC
keywords pickupanddeliveryproblemtwo-dimensionalpackingfeasibilityinferencedominance-basedcheckingvehicleroutingconstraintsorder-preservingmappingno-relocation
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 introduces a dominance-based feasibility framework for pickup and delivery problems that include two-dimensional packing constraints. Items loaded and unloaded along the route cause the available space to change non-monotonically, so repeated exact packing checks become the main computational cost. The method defines an order-preserving mapping that records both geometric containment of items and the sequence of pickups and deliveries; this mapping decides whether a candidate plan is feasible by embedding it into an already-verified reference plan. When dominance holds, the framework skips the expensive exact packing routine. Tests show the approach cuts total feasibility-checking time by up to 42 percent by lowering the number of exact packing calls.

What carries the argument

The order-preserving mapping, which embeds a candidate packing state into a verified reference plan by matching both item geometry and pickup-delivery order.

What would settle it

A concrete counter-example in which the mapping declares a feasible plan infeasible, or an infeasible plan feasible, would show the inference rule is unsafe.

Watch

Extended reading notes

Core claim

Under no-relocation constraints, an order-preserving mapping that jointly captures geometric containment and sequence compatibility enables dominance-based inference: feasibility of a new packing state can be safely decided by reference to a previously verified plan, without missing valid solutions or producing false positives.

Load-bearing premise

An order-preserving mapping under no-relocation constraints can always correctly decide feasibility without false positives or missed solutions.

Editorial extensions

If this is right

  • Exact packing procedures are called only when dominance cannot be established.
  • The framework can be inserted into both exact and heuristic routing solvers.
  • Three search rules and storage strategies reduce the cost of screening candidate reference plans.
  • Overall feasibility verification time falls because effort shifts away from the slowest stage.

Reading between the lines

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

  • The same dominance idea could be tested on problems that add time windows or three-dimensional loading.
  • Storing verified states by sequence signature might generalize to other order-dependent resource problems.
  • If the mapping overhead stays low, larger route sets become solvable before the exact packer dominates runtime.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 1 minor

Summary. The paper introduces a dominance-based feasibility inference framework for the two-dimensional pickup and delivery problem with packing constraints (2P-PDP). Under no-relocation constraints, it defines an order-preserving mapping that jointly encodes geometric containment and sequence compatibility to infer feasibility of new packing states from verified reference plans, thereby reducing calls to exact packing procedures. Computational experiments are reported to show up to 42% reduction in feasibility-checking time relative to a benchmark without dominance.

Significance. If the order-preserving mapping is sound, the framework offers a generic, embeddable technique that could meaningfully accelerate both exact and heuristic solvers for routing problems with dynamic, non-monotonic packing constraints—a recognized bottleneck. The reported time savings arise from shifting verification effort away from the most expensive stage, which would be a practical contribution if the completeness and soundness properties hold.

major comments (1)
  1. [abstract / framework description] The central claim that the order-preserving mapping safely infers feasibility (no false positives) and completeness (no missed solutions) rests on an assertion that it jointly captures geometric containment and sequence compatibility under no-relocation constraints. No derivation, invariant, or counter-example analysis is supplied showing why this suffices when free-space evolution is non-monotonic; without such justification the inference step could accept invalid packings or reject valid ones.
minor comments (1)
  1. [abstract] The abstract states a 42% time reduction but supplies no information on experimental design, instance sets, baselines, statistical significance, or post-hoc selection; the full manuscript should include these details to allow verification of the computational claims.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and for highlighting the need for explicit justification of the order-preserving mapping. We address the single major comment below and will incorporate the requested material in the revision.

read point-by-point responses

  1. Referee: [abstract / framework description] The central claim that the order-preserving mapping safely infers feasibility (no false positives) and completeness (no missed solutions) rests on an assertion that it jointly captures geometric containment and sequence compatibility under no-relocation constraints. No derivation, invariant, or counter-example analysis is supplied showing why this suffices when free-space evolution is non-monotonic; without such justification the inference step could accept invalid packings or reject valid ones.

    Authors: We agree that the current manuscript lacks a self-contained derivation or invariant argument establishing soundness and completeness of the order-preserving mapping when free-space evolution is non-monotonic. In the revised version we will add a dedicated subsection (new Section 3.2) that (i) states the invariant preserved by the mapping under the no-relocation constraint, (ii) proves that joint encoding of geometric containment and sequence compatibility is sufficient to guarantee both no false positives and no missed feasible states, and (iii) supplies a short counter-example analysis showing why non-monotonicity does not break the inference when order is preserved. This addition directly addresses the concern without altering the algorithmic framework or experimental claims. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: algorithmic proposal with empirical validation

full rationale

The paper introduces an order-preserving mapping as a new construct to enable dominance-based feasibility inference under no-relocation constraints. This mapping is defined to jointly capture geometric containment and sequence compatibility, then embedded into verified reference plans. The 42% time reduction is reported from computational experiments that count fewer exact packing calls, not from any fitted parameter or self-referential definition. No load-bearing step reduces by construction to its own inputs, and no self-citation chain is invoked to justify the core inference rule. The derivation chain is self-contained as an algorithmic contribution.

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

Abstract-only review provides no explicit free parameters, axioms, or invented entities; the framework appears to rely on standard algorithmic assumptions about dominance and order preservation that are not detailed here.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Dominance-Based Feasibility Inference for Packing-Constrained Pickup and Delivery Problems." pith.science (2026). https://pith.science/paper/36ULCR2U

@misc{pith2026260625970,
  author       = {Pith},
  title        = {Pith review of: Dominance-Based Feasibility Inference for Packing-Constrained Pickup and Delivery Problems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/36ULCR2U}},
  note         = {Machine review of arXiv:2606.25970}
}
read the original abstract

Routing and packing are intrinsically coupled in transport problems, requiring joint planning for cost-efficient and physically realizable solutions. We study a pickup and delivery problem with two-dimensional packing constraints (2P-PDP). Unlike vehicle routing variants where items are loaded before vehicles leave the depot and packing is validated only once, the 2P-PDP induces non-monotonic free-space evolution, substantially increasing feasibility-checking complexity. To address this bottleneck, we propose a generic dominance-based feasibility framework that is embeddable in a broad class of exact and heuristic routing algorithms. Under no-relocation constraints, inferring feasibility from a previously verified packing state requires preserving the pickup and delivery order of onboard items. To this end, we introduce an order-preserving mapping that jointly captures geometric containment and sequence compatibility, enabling dominance-based inference by embedding the new packing state into a verified reference plan. To further reduce dominance-screening overhead, we design three search rules to guide candidate exploration and tailored strategies to store, retrieve, and prioritize verified states. Computational experiments show that the proposed approach reduces feasibility-checking time by up to 42% compared to a benchmark without dominance. The improvement stems from reducing exact packing-procedure calls, shifting verification effort away from the most computationally expensive stage.

Figures

Figures reproduced from arXiv: 2606.25970 by the authors.

Figure 1
Figure 1. Illustration of the no-relocation requirement under the vehicle coordinate system. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. An illustrative route and its SOPPs The above definitions localize packing feasibility to a finite set of critical packing states. Proposition 1 connects these local checks to route-level packing feasibility. Proposition 1 (Route packing feasibility via sequential SOPP verification). Let κ = (n0, n1, . . . , nm) be a route and let A(κ) = (a0, a1, . . . , am−1), where aℓ = (nℓ , nℓ+1) for ℓ = 0, . . . , m − 1, denote… view at source ↗
Figure 3
Figure 3. Bin–piece interpretation for dominance checking under separate NR-p and NR-d [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (7 more)
Figure 4
Figure 4. Figure 4: Illustration of partial dominance under the NR-d condition and its extension to full [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Illustration of the OPM search tree and the associated pruning rules. Panel (a) shows [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Median improvement over the benchmark across item-size and trunk-width scaling. [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]
Figure 7
Figure 7. Figure 7: Median improvement over the benchmark across item shape regularization ratio [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: Outperformance probability and CP-stage verification proportion across time [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]
Figure 9
Figure 9. Figure 9: Median and IQR of the objective value over the ( [PITH_FULL_IMAGE:figures/full_fig_p024_9.png]
Figure 10
Figure 10. Figure 10: Median and IQR of runtime over the (µ candi, µentry) grid 628 seconds. The setting (30, 60) achieves a comparable median runtime of 648 seconds, while reducing the runtime IQR from 277 to 136 seconds, corresponding to a reduction of about 51.0%. This indicates better …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 2 canonical work pages

  1. [1]

    Computers & Operations Research , pages=

    A branch-and-cut algorithm for the pallet-loading vehicle routing problem considering load balance of semi-trailer trucks , author=. Computers & Operations Research , pages=. 2025 , publisher=

    2025

  2. [2]

    European Journal of Operational Research , volume=

    A branch-and-cut algorithm for the vehicle routing problem with two-dimensional loading constraints , author=. European Journal of Operational Research , volume=. 2022 , publisher=

    2022

  3. [3]

    arXiv preprint arXiv:2402.14868 , year=

    A branch-and-cut algorithm for vehicle routing problems with three-dimensional loading constraints , author=. arXiv preprint arXiv:2402.14868 , year=

    work page arXiv

  4. [4]

    Expert Systems with Applications , volume=

    A branch-and-cut approach for the vehicle routing problem with loading constraints , author=. Expert Systems with Applications , volume=. 2016 , publisher=

    2016

  5. [5]

    Transportation Science , volume=

    A branch-and-price-and-cut algorithm for the vehicle routing problem with two-dimensional loading constraints , author=. Transportation Science , volume=. 2022 , publisher=

    2022

  6. [6]

    International Transactions in Operational Research , volume=

    A branch-and-price-based heuristic for the vehicle routing problem with two-dimensional loading constraints and time windows , author=. International Transactions in Operational Research , volume=. 2024 , publisher=

    2024

  7. [7]

    International Journal of Production Research , volume=

    A column generation based heuristic for the capacitated vehicle routing problem with three-dimensional loading constraints , author=. International Journal of Production Research , volume=. 2017 , publisher=

    2017

  8. [8]

    European Journal of Operational Research , volume=

    A hybrid algorithm for the vehicle routing problem with pickup and delivery and three-dimensional loading constraints , author=. European Journal of Operational Research , volume=. 2016 , publisher=

    2016

Show all 50 references
  1. [9]

    Mathematics , volume=

    A matheuristic approach to the integration of three-dimensional bin packing problem and vehicle routing problem with simultaneous delivery and pickup , author=. Mathematics , volume=. 2023 , publisher=

    2023

  2. [10]

    Applied Soft Computing , volume=

    A tabu search based approach for the heterogeneous fleet vehicle routing problem with three-dimensional loading constraints , author=. Applied Soft Computing , volume=. 2022 , publisher=

    2022

  3. [11]

    Transportation Science , volume=

    An exact approach for the vehicle routing problem with two-dimensional loading constraints , author=. Transportation Science , volume=. 2007 , publisher=

    2007

  4. [12]

    Transportation Research Part E: Logistics and Transportation Review , volume=

    Collaborative multidepot split delivery network design with three-dimensional loading constraints , author=. Transportation Research Part E: Logistics and Transportation Review , volume=. 2025 , publisher=

    2025

  5. [13]

    Computers & Industrial Engineering , volume=

    Constraint programming-based solution approaches for three-dimensional loading capacitated vehicle routing problems , author=. Computers & Industrial Engineering , volume=. 2022 , publisher=

    2022

  6. [14]

    Computers & Operations Research , volume=

    Effective loading in combined vehicle routing and container loading problems , author=. Computers & Operations Research , volume=. 2023 , publisher=

    2023

  7. [15]

    European Journal of Operational Research , volume=

    Heuristics for vehicle routing problems with backhauls, time windows, and 3D loading constraints , author=. European Journal of Operational Research , volume=. 2018 , publisher=

    2018

  8. [16]

    Transportation Research Part E: Logistics and Transportation Review , volume=

    Pickup capacitated vehicle routing problem with three-dimensional loading constraints: Model and algorithms , author=. Transportation Research Part E: Logistics and Transportation Review , volume=. 2023 , publisher=

    2023

  9. [17]

    Computers & Operations Research , volume=

    Solving capacitated vehicle routing problem with three-dimensional loading and relocation constraints , author=. Computers & Operations Research , volume=. 2025 , publisher=

    2025

  10. [18]

    European Journal of Operational Research , volume=

    Solving the pickup and delivery problem with three-dimensional loading constraints and reloading ban , author=. European Journal of Operational Research , volume=. 2018 , publisher=

    2018

  11. [19]

    Transportation Science , volume=

    The pallet-packing vehicle routing problem , author=. Transportation Science , volume=. 2012 , publisher=

    2012

  12. [20]

    European Journal of Operational Research , volume=

    The split delivery vehicle routing problem with three-dimensional loading constraints , author=. European Journal of Operational Research , volume=. 2020 , publisher=

    2020

  13. [21]

    European Journal of Operational Research , volume=

    The split heterogeneous vehicle routing problem with three-dimensional loading constraints on a large scale , author=. European Journal of Operational Research , volume=. 2022 , publisher=

    2022

  14. [22]

    European Journal of Operational Research , volume=

    The vehicle routing problem with simultaneous pick-ups and deliveries and two-dimensional loading constraints , author=. European Journal of Operational Research , volume=. 2016 , publisher=

    2016

  15. [23]

    Computers & Operations Research , volume=

    A hybrid algorithm for the capacitated vehicle routing problem with three-dimensional loading constraints , author=. Computers & Operations Research , volume=. 2012 , publisher=

    2012

  16. [24]

    Informs Journal on computing , volume=

    Extreme point-based heuristics for three-dimensional bin packing , author=. Informs Journal on computing , volume=. 2008 , publisher=

    2008

  17. [25]

    International Transactions in Operational Research , volume=

    Variable neighborhood search algorithms for the vehicle routing problem with two-dimensional loading constraints and mixed linehauls and backhauls , author=. International Transactions in Operational Research , volume=. 2020 , publisher=

    2020

  18. [26]

    European Journal of Operational Research , volume=

    Hybrid algorithms for the vehicle routing problem with clustered backhauls and 3D loading constraints , author=. European Journal of Operational Research , volume=. 2015 , publisher=

    2015

  19. [27]

    International Conference on Computational Logistics , pages=

    A branch-and-price algorithm for the vehicle routing problem with 2-dimensional loading constraints , author=. International Conference on Computational Logistics , pages=. 2016 , organization=

    2016

  20. [28]

    1988 , institution =

    An Algorithm for the Shortest Path Problem with Resource Constraints , author =. 1988 , institution =

    1988

  21. [29]

    Networks: An International Journal , volume=

    An exact algorithm for the elementary shortest path problem with resource constraints: Application to some vehicle routing problems , author=. Networks: An International Journal , volume=. 2004 , publisher=

    2004

  22. [30]

    Operations Research , volume=

    An exact algorithm for the two-dimensional orthogonal packing problem with unloading constraints , author=. Operations Research , volume=. 2014 , publisher=

    2014

  23. [31]

    Networks , year=

    Exact Methods for the Split Delivery Vehicle Routing Problem With Two-Dimensional Loading Constraints , author=. Networks , year=

  24. [32]

    arXiv preprint arXiv:2406.12454 , year=

    A neural column generation approach to the vehicle routing problem with two-dimensional loading and last-in-first-out constraints , author=. arXiv preprint arXiv:2406.12454 , year=

  25. [33]

    Column Generation Algorithms for the Pickup and Delivery Problem , author=. Beyond Frontiers of Operations Research: Emerging Technologies and Innovative Optimization Paradigms: Proceedings of the 7th AIROYoung Workshop (February 15-17, 2023, University of Milan, Italy) and 8t...

    2023

  26. [34]

    Computers & Operations Research , volume=

    An improved skyline based heuristic for the 2D strip packing problem and its efficient implementation , author=. Computers & Operations Research , volume=. 2017 , publisher=

    2017

  27. [35]

    Operations Research , volume=

    A new placement heuristic for the orthogonal stock-cutting problem , author=. Operations Research , volume=. 2004 , publisher=

    2004

  28. [36]

    Applied Mathematical Modelling , volume=

    An open space based heuristic for the 2D strip packing problem with unloading constraints , author=. Applied Mathematical Modelling , volume=. 2019 , publisher=

    2019

  29. [37]

    Computers & Industrial Engineering , volume=

    Developing a simulated annealing algorithm for the cutting stock problem , author=. Computers & Industrial Engineering , volume=. 1997 , publisher=

    1997

  30. [38]

    European journal of operational research , volume=

    A simulated annealing algorithm for the capacitated vehicle routing problem with two-dimensional loading constraints , author=. European journal of operational research , volume=. 2018 , publisher=

    2018

  31. [39]

    European Journal of Operational Research , volume=

    A variable neighborhood search for the capacitated vehicle routing problem with two-dimensional loading constraints , author=. European Journal of Operational Research , volume=. 2015 , publisher=

    2015

  32. [40]

    European Journal of Operational Research , volume=

    A variable neighborhood search for the green vehicle routing problem with two-dimensional loading constraints and split delivery , author=. European Journal of Operational Research , volume=. 2024 , publisher=

    2024

  33. [41]

    Computers & Operations Research , volume=

    An exact approach for the green vehicle routing problem with two-dimensional loading constraints and split delivery , author=. Computers & Operations Research , volume=. 2021 , publisher=

    2021

  34. [42]

    Computers & Operations Research , volume=

    A new constraint programming approach for the orthogonal packing problem , author=. Computers & Operations Research , volume=. 2008 , publisher=

    2008

  35. [43]

    INFORMS Journal on Computing , volume=

    Using decomposition techniques and constraint programming for solving the two-dimensional bin-packing problem , author=. INFORMS Journal on Computing , volume=. 2007 , publisher=

    2007

  36. [44]

    Transportation science , volume=

    An adaptive large neighborhood search heuristic for the pickup and delivery problem with time windows , author=. Transportation science , volume=. 2006 , publisher=

    2006

  37. [45]

    Working Paper Series , year=

    Hybrid Algorithms for the Vehicle Routing Problem with Pickup and Delivery and Two-dimensional Loading Constraints , author=. Working Paper Series , year=

  38. [46]

    2024 , howpublished =

    2024

  39. [47]

    2L-PDP-instances: Benchmark instances for the Vehicle Routing Problem with Pickup and Delivery and Two-dimensional Loading Constraints , year =

    M. 2L-PDP-instances: Benchmark instances for the Vehicle Routing Problem with Pickup and Delivery and Two-dimensional Loading Constraints , year =

  40. [48]

    Biometrics Bulletin , volume =

    Wilcoxon, Frank , title =. Biometrics Bulletin , volume =

  41. [49]

    Conover, W. J. , title =

  42. [50]

    A Tabu Search Algorithm for a Routing and Container Loading Problem , volume =

    Michel Gendreau and Manuel Iori and Gilbert Laporte and Silvano Martello , journal =. A Tabu Search Algorithm for a Routing and Container Loading Problem , volume =

Pith tools

Reviewed June 25, 2026 · model on record in the stance chip above.