Graph-Coarsening Approach for the Capacitated Vehicle Routing Problem with Time Windows

Mustafa Mert \"Ozy{\i}lmaz

arxiv: 2510.22329 · v2 · submitted 2025-10-25 · 💻 cs.AI · math.OC

Graph-Coarsening Approach for the Capacitated Vehicle Routing Problem with Time Windows

Mustafa Mert \"Ozy{\i}lmaz This is my paper

Pith reviewed 2026-05-18 04:11 UTC · model grok-4.3

classification 💻 cs.AI math.OC

keywords CVRPTWgraph coarseningvehicle routingtime windowsquantum annealingSolomon benchmarksmultilevel optimizationheuristics

0 comments

The pith

Graph coarsening aggregates customers into meta-nodes to solve CVRPTW faster with classical heuristics and quantum annealing.

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

The paper develops a multilevel coarsening strategy for the Capacitated Vehicle Routing Problem with Time Windows that groups customers using a combined spatial and temporal distance. The smaller coarsened instance is solved by standard heuristics or by quantum annealing hardware and the solution is then lifted back to the original customers with arrival times recalculated. Experiments on the full set of Solomon benchmark problems show clear reductions in running time for classical methods while solution quality stays comparable. On small instances solved by quantum methods the approach also trims vehicle counts and total route duration for clustered customer sets and keeps every solution feasible.

Core claim

A multilevel graph coarsening and refinement procedure that forms meta-nodes according to a spatio-temporal distance metric allows the CVRPTW to be solved on a reduced graph by classical heuristics or quantum annealing; the resulting routes are expanded to the original customer set by recomputing arrival times and recording any constraint violations, yielding substantially lower computation times while preserving or improving solution quality on clustered instances.

What carries the argument

Multilevel coarsening that aggregates customers into meta-nodes via a spatio-temporal distance metric, followed by refinement that recomputes arrival times on the original graph.

If this is right

Classical heuristics run in significantly less time on Solomon benchmarks with little or no drop in solution quality.
Quantum annealing on all 56 Solomon instances at five and ten customers finishes faster after coarsening.
Clustered (C-type) instances show simultaneous drops in vehicle count and route duration while remaining fully feasible.
Narrow-window random (R-type) instances limit how deeply coarsening can be applied before feasibility drops.

Where Pith is reading between the lines

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

The same coarsening step could be inserted as a preprocessing stage for any existing CVRPTW solver to handle larger customer sets.
Instance-type classification before coarsening might further improve results by skipping the step on hard random cases.
The approach suggests a general pattern for other constrained routing problems where a distance metric can capture both geometry and scheduling limits.

Load-bearing premise

The spatio-temporal distance used to create meta-nodes keeps enough of the original routing structure that the refinement step can recover feasible, high-quality solutions without systematic bias.

What would settle it

If refinement after coarsening produces solutions that violate time windows or capacity limits on most clustered Solomon instances, or yields worse vehicle counts and durations than uncoarsened solves, the performance claims would not hold.

Figures

Figures reproduced from arXiv: 2510.22329 by Mustafa Mert \"Ozy{\i}lmaz.

**Figure 2.** Figure 2: Overall workflow of the proposed spatio–temporal graph [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Figure 3: A subset of instances of the Solomon [17] dataset used for [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 6.** Figure 6: Comparison of total travel distance for Greedy and Savings [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗

**Figure 7.** Figure 7: Total route duration comparison for coarsened and inflated [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗

**Figure 5.** Figure 5: Distribution of time window violations for coarsened and [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗

**Figure 10.** Figure 10: Example of an uncoarsened solution [PITH_FULL_IMAGE:figures/full_fig_p008_10.png] view at source ↗

**Figure 11.** Figure 11: Solution after coarsening procedure P (Target Node Percentage): Results show a clear trade-off between coarsening level and solution quality. Higher P values (less coarsening) lead to better median objective scores but smaller reductions in graph size. Radius Coefficient (Rcoef f ): Results indicate a sweet spot for this parameter. Too small yields minimal coarsening; too large produces poor merges. Valu… view at source ↗

**Figure 13.** Figure 13: Distribution of total inflated distance across all experiments [PITH_FULL_IMAGE:figures/full_fig_p009_13.png] view at source ↗

**Figure 14.** Figure 14: Distribution of total inflated distance for different [PITH_FULL_IMAGE:figures/full_fig_p009_14.png] view at source ↗

**Figure 15.** Figure 15: Total inflated distance for varying probability thresholds [PITH_FULL_IMAGE:figures/full_fig_p010_15.png] view at source ↗

**Figure 16.** Figure 16: Distribution of total inflated distance as a function of [PITH_FULL_IMAGE:figures/full_fig_p010_16.png] view at source ↗

**Figure 17.** Figure 17: Exemplary results of Savings solver for C101 instance and [PITH_FULL_IMAGE:figures/full_fig_p011_17.png] view at source ↗

**Figure 19.** Figure 19: Average computation time comparison between Uncoarsened [PITH_FULL_IMAGE:figures/full_fig_p013_19.png] view at source ↗

read the original abstract

The Capacitated Vehicle Routing Problem with Time Windows (CVRPTW) is a fundamental NP-hard optimization problem in logistics. Solving large-scale instances remains computationally challenging for exact solvers. This paper introduces a multilevel graph coarsening and refinement strategy that aggregates customers into meta-nodes based on a spatio-temporal distance metric. The reduced problem is solved using both classical heuristics and quantum annealing hardware, then expanded back into the original space with arrival times recomputed and constraint violations recorded. Comprehensive experiments on Solomon benchmarks demonstrate that our method significantly reduces computation time while preserving solution quality for classical heuristics. For quantum solvers, experiments across all 56 Solomon instances at $N=5$ and $N=10$ customers show that coarsening consistently reduces computation time and, on clustered (C-type) instances, simultaneously reduces vehicle count and route duration with no feasibility loss. Coarsening effectiveness is strongly instance-structure dependent: C-type instances achieve %100 post-coarsening feasibility with measurable quality improvements, while narrow-window random (R-type) instances present structural constraints that limit achievable coarsening depth.

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.

Desk Editor's Note private letter to a colleague

The paper shows time savings from spatio-temporal coarsening on small CVRPTW instances for both classical and quantum solvers, but the refinement step and scale limits keep the practical payoff modest.

read the letter

The key takeaway is that this coarsening method with a custom spatio-temporal metric speeds up solves on tiny CVRPTW instances and delivers measurable gains in vehicle count and duration for clustered Solomon cases when paired with quantum annealing, all while claiming no feasibility loss on those types. It is a straightforward multilevel reduction applied to a standard logistics problem rather than a broad new framework. What the work does well is run the approach across the full set of 56 Solomon instances at N=5 and N=10, report consistent time reductions, and explicitly note how instance structure (C-type versus narrow-window R-type) affects how much coarsening is possible. That dependence is called out plainly, which is useful. The quantum experiments are a concrete extension even if they stay small. The soft spots are around the refinement process and the limited scope. The central premise is that the metric lets feasible coarsened solutions expand back without introducing hidden violations or big quality drops, yet the description gives no counts on recovery rates, no worst-case bounds, and no breakdown of how much quality is lost when the metric falls short on R-type cases. Quantum results are confined to N=5 and 10, which is too small to speak to the large-scale regime the introduction targets. Without tables or direct baseline comparisons in the summary it is hard to gauge how much the reported improvements exceed standard reduction tricks. This is aimed at researchers working on heuristic or hybrid quantum solvers for vehicle routing who want benchmarked reduction ideas. It has enough concrete experiments on standard data to deserve a serious referee, even though the gains look incremental and the feasibility details need tightening.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces a multilevel graph coarsening and refinement strategy for the Capacitated Vehicle Routing Problem with Time Windows (CVRPTW). Customers are aggregated into meta-nodes via a spatio-temporal distance metric; the reduced instance is solved using classical heuristics or quantum annealing (restricted to N=5 and N=10), then expanded back to the original space with arrival times recomputed and constraint violations recorded. Experiments on Solomon benchmarks claim that the method reduces computation time while preserving solution quality for classical heuristics, and for quantum solvers on all 56 instances at small N it reduces time and, on C-type instances, simultaneously reduces vehicle count and route duration with no feasibility loss. Effectiveness is strongly dependent on instance structure, with 100% post-coarsening feasibility on C-type but structural limits on narrow-window R-type instances.

Significance. If the spatio-temporal metric reliably preserves enough structure for the refinement step to recover feasible high-quality solutions without systematic bias, the approach could provide a practical route to scaling CVRPTW solvers, especially for quantum hardware limited to small problem sizes. The instance-structure dependence and explicit recording of violations are positive features. However, the current evidence base is narrow (quantum results only at N=5/10, no quantitative tables or baseline timings), so the significance for large-scale classical or quantum deployment remains provisional pending stronger empirical support.

major comments (2)

[Experiments] Experiments section (and abstract): the claims of 'significantly reduces computation time while preserving solution quality' and 'reduces vehicle count and route duration with no feasibility loss' on Solomon instances are unsupported by any tables, exact timings, baseline comparisons, standard deviations, or counts of feasibility violations recovered. Without these data the magnitude and reliability of the reported gains cannot be evaluated.
[Section 3] Refinement procedure (Section 3): the central premise that the spatio-temporal distance metric preserves time-window structure sufficiently for any feasible coarsened solution to refine to a feasible original solution is invoked but not characterized. No weighting details (time-window overlap vs. spatial distance), no quantitative bound on feasibility recovery rate, and no worst-case analysis for R-type instances are provided, even though the abstract notes that R-type instances limit coarsening depth.

minor comments (1)

[Abstract] Abstract: '%100 post-coarsening feasibility' is a typographical error and should read '100%'.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review. We address each major comment below and have revised the manuscript to strengthen the empirical support and technical characterization as suggested.

read point-by-point responses

Referee: [Experiments] Experiments section (and abstract): the claims of 'significantly reduces computation time while preserving solution quality' and 'reduces vehicle count and route duration with no feasibility loss' on Solomon instances are unsupported by any tables, exact timings, baseline comparisons, standard deviations, or counts of feasibility violations recovered. Without these data the magnitude and reliability of the reported gains cannot be evaluated.

Authors: We agree that the current presentation of results relies on summary claims without sufficient quantitative backing. In the revised manuscript we will add comprehensive tables for both the classical heuristic and quantum annealing experiments. These will report exact wall-clock times, direct baseline comparisons against the same solvers run on the original (uncoarsened) instances, standard deviations over repeated runs, and explicit counts of feasibility violations recovered during refinement. This will enable readers to assess the magnitude, consistency, and reliability of the reported improvements. revision: yes
Referee: [Section 3] Refinement procedure (Section 3): the central premise that the spatio-temporal distance metric preserves time-window structure sufficiently for any feasible coarsened solution to refine to a feasible original solution is invoked but not characterized. No weighting details (time-window overlap vs. spatial distance), no quantitative bound on feasibility recovery rate, and no worst-case analysis for R-type instances are provided, even though the abstract notes that R-type instances limit coarsening depth.

Authors: We accept that the refinement step requires explicit characterization. The revised Section 3 will specify the exact weighting formula balancing time-window overlap against spatial distance, report empirical feasibility recovery rates across the Solomon benchmark classes, and include a dedicated discussion of worst-case behavior for R-type instances, explaining the structural constraints that limit coarsening depth when time windows are narrow. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical results rely on external Solomon benchmarks and standard solvers

full rationale

The paper describes a graph-coarsening approach that aggregates customers into meta-nodes via a spatio-temporal distance metric, solves the reduced CVRPTW instance using classical heuristics or quantum annealing, and refines solutions back to the original space while recording violations. All reported performance gains (reduced computation time, preserved or improved quality on C-type instances) are obtained from experiments on the fixed, external Solomon benchmark set across 56 instances at small N. No equations, fitted parameters, or self-citations are presented that would make any prediction or quality metric equivalent to the method's own inputs by construction. The derivation chain is therefore self-contained against independent external data and solvers.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the unproven effectiveness of the chosen spatio-temporal distance for aggregation and on the assumption that post-refinement constraint repair does not degrade solution quality in a systematic way. No free parameters are explicitly named in the abstract, but coarsening depth and metric weights are implicit choices.

free parameters (1)

coarsening depth / aggregation threshold
Controls how many customers are merged into each meta-node; must be selected to balance size reduction against feasibility recovery.

axioms (1)

domain assumption Spatio-temporal distance metric groups customers such that feasible routes on the coarsened graph can be expanded to feasible routes on the original graph
Invoked when the paper states that the reduced problem is solved and then expanded back with arrival times recomputed.

pith-pipeline@v0.9.0 · 5718 in / 1433 out tokens · 44076 ms · 2026-05-18T04:11:59.514346+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Dij =α·τij +β·∆Tij ... feasibility checks ... aggregated time window [e′,ℓ′]
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean alpha_pin_under_high_calibration unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

radiusCoeff ... P(Target percentage of nodes remaining) ... random search over {0.1,0.5,0.9}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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Polynomial-time algorithms for prime factoriza- tion and discrete logarithms on a quantum computer,

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Quantum annealing in the transverse ising model,

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M. Qi, G. Ding, Y. Zhou, and L. Miao, “Vehicle routing problem with time windows based on spatiotemporal distance,”Journal of Transportation Systems Engineering and Information Tech- nology, vol. 11, no. 1, pp. 36–42, 2011

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M. Desrochers, J. Desrosiers, and M. Solomon, “A new opti- mization algorithm for the vehicle routing problem with time windows,”Operations Research, vol. 40, pp. 342–354, 04 1992

work page 1992

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work page arXiv 2018

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work page 2024

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work page 1987

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G. Chen, J. Gao, and D. Chen, “Research on vehicle routing problem with time windows based on improved genetic algorithm and ant colony algorithm,”Electronics, vol. 14, no. 4,

work page

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Available: https://www.mdpi.com/2079-9292/ 14/4/647

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Z. Liu, W. Wang, J. He, J. Zhang, J. Wang, S. Li, Y. Sun, and X. Ren, “A new hybrid algorithm for vehicle routing optimization,”Sustainability, vol. 15, no. 14, 2023. [Online]. Available: https://www.mdpi.com/2071-1050/15/14/10982

work page 2023

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Stas crossover with k-mean clustering for vehicle routing problem with time window,

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work page

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Time-dependent vehicle rout- ing problem in subway-assisted delivery systems,

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