{"id":"ce310268-873c-4329-9e4a-8149aac8b103","arxiv_id":"2412.02720","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"H2S and H3S, two fuzzy-clustering plus quantum-annealing hybrids, achieve 8-19 percent optimality gaps on five small VRPLib instances, with H3S favored on corner-depot instances and H2S on center-depot instances.","lead":"Two hybrid quantum-classical algorithms for the Capacitated Vehicle Routing Problem combine fuzzy clustering on a classical computer with quantum annealing for routing. The paper reports solutions within 8 to 19 percent of known optima on five small benchmark instances, but provides no code, no error bars, and no classical baseline.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"QUBO encodings in §4.4.2 are not printed, so feasibility of the reported routes cannot be verified; the paper itself reports an infeasible H2S run on A-n33-k6.","rationale":"The reader's weakest assumption is that the QUBO Hamiltonians in §4.4.2 correctly encode the CVRP constraints, and that the D-Wave output is therefore a feasible route. This is exactly the most load-bearing concern: the entire comparison against best-known solutions in Table 2 collapses if the reported tours violate capacity or visit constraints. The manuscript does not print the full QUBO equations, provides no code, and reports one H2S instance with no feasible solution, which is direct evidence that infeasibility occurs in the pipeline. The reader's REJECT verdict is justified by this unverifiability, independent of performance claims. I agree with the reader's identification of the weakest assumption. My concrete test would settle the matter by checking whether the QUBO, when solved, yields feasible routes; if it does, the central claim could be re-evaluated, but until then the paper does not substantiate 'promising results'. I am not raising a new concern about performance baselines or novelty, because the correctness of the optimization model is logically prior to any cost or time comparison. An honest non-finding is not appropriate here: the garbled equations and the reported infeasible run are concrete, textually present red flags, not manufactured objections.","tokens_in":12375,"tokens_out":2267,"duration_ms":23183,"concrete_test":"Obtain or reconstruct the QUBO from the authors' code or Leap workspace (or re-derive it from a corrected §4.4.2), then for each of the five Table 2 instances solve the QUBO classically with simulated annealing or exact enumeration. Decode the lowest-energy bitstrings into routes and check all CVRP constraints: each customer visited exactly once, each truck leaves the depot once, capacity is not exceeded, and no subtours exist. If any decoded optimum is infeasible, or if the recovered Hamiltonian differs from the paper's constraint statements, the reported costs cannot be regarded as valid CVRP solutions.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that H2S and H3S produce 'promising results' on five CVRP instances (Table 2). This requires that every reported route is a feasible CVRP tour, respecting single-visit, depot-first, route-validity, capacity, and subtour-elimination constraints. However, §4.4.2 never displays the actual QUBO Hamiltonians: the objective function and all five Hamiltonian equations are garbled or missing in the manuscript. A reader cannot verify that the D-Wave output is decoded into tours satisfying these constraints. This is not a cosmetic gap: the paper itself states 'There was no feasible solution found for the H2S instance of A-n33-k6,' showing that the pipeline can return infeasible output. If the capacity penalty is mis-summed, or if the auxiliary 'sequence of steps' variable is not correctly mapped to the edge variables, then the reported costs 855, 873, 874, 896, 766 are costs of possibly invalid routes, not comparable to the best-known feasible CVRP values (784, 661, 778, 799, 669). Since no code, QUBO matrices, or D-Wave sample logs are provided, the feasibility of every reported solution is unverified, so the central claim is not substantiated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes two hybrid classical-quantum strategies, H2S and H3S, for the Capacitated Vehicle Routing Problem (CVRP). Both use Fuzzy C-Means clustering on a classical computer and quantum annealing on a D-Wave machine for routing: H2S solves a TSP per cluster, while H3S first solves a CVRP over cluster centroids and then a TSP per route. The authors evaluate the strategies on five small VRPLib instances (32–37 customers) and report costs with optimality gaps of roughly 8–19% relative to best-known solutions. They also offer a preliminary, qualitative analysis of when each approach performs better based on customer distribution and depot location.","tokens_in":12703,"tokens_out":5303,"duration_ms":45089,"significance":"If the reported solutions are feasible and the encoding is correct, the paper demonstrates that a hybrid fuzzy-clustering/quantum-annealing pipeline can produce moderate-quality CVRP tours on small instances. This would be a useful application note for practitioners exploring hybrid quantum-classical optimization. However, the current manuscript does not provide the actual QUBO Hamiltonians, does not report repeated runs or statistical measures, and lacks any classical baseline, so the central performance claim is not yet substantiated. The contribution is potentially salvageable but requires substantial additional evidence.","major_comments":[{"comment":"The objective function and all five Hamiltonian equations for the CVRP QUBO are missing or garbled in the manuscript; the formulas do not render. Since the central claim that the reported routes are feasible CVRP tours depends entirely on this QUBO correctly encoding single-visit, depot-first, route-validity, capacity, and subtour-elimination constraints, the reader cannot verify that the costs in Table 2 correspond to valid solutions. The paper's own statement that 'There was no feasible solution found for the H2S instance of A-n33-k6' shows the pipeline can produce infeasible output, and without the explicit Hamiltonians, penalty weights, and decoding procedure, the feasibility of every reported solution is unverified.","section":"Section 4.4.2"},{"comment":"The text immediately after Table 2 claims that 'the H3S solutions are significantly lesser than the H2S solutions for instances A–n33-k5, A-n33-k6, A-n34-k5, A-n37-k5', but the table shows the opposite for A-n33-k5 (H2S: 695 vs H3S: 712) and A-n37-k5 (H2S: 763 vs H3S: 766), and A-n33-k6 does not appear in Table 2 at all. The comparative evaluation of H2S and H3S is therefore contradicted by the paper's own data.","section":"Section 5, Table 2 and text"},{"comment":"The experimental methodology reports only a single run per instance, with no error bars, no repeated annealing samples, and no statistical analysis. Since D-Wave annealing is stochastic, a single solution value is insufficient to support the optimality-gap claims. Moreover, no classical baseline is provided (e.g., the same clustering followed by a classical TSP solver, or a standard CVRP heuristic), so the abstract's claim of 'promising results, both in terms of solution time and solution cost' is unsupported: no comparison of solution time to classical alternatives is given, and the absolute gaps of 8–19% are not contextualized.","section":"Section 5"}],"minor_comments":[{"comment":"The FCM centroid and membership-update equations are not rendered; please check the mathematical display.","section":"Section 4.1"},{"comment":"The sentence 'We begin the assignment algorithm by associating each node with its most preferred cluster centroid, .' contains a missing symbol and should be completed.","section":"Section 4.2"},{"comment":"The text refers to 'From table 1' when discussing the results table; the table numbering should be corrected to Table 2.","section":"Section 5"},{"comment":"Figure numbering is inconsistent: the text references Fig. 4 and Fig. 5, but the displayed figures are labeled Fig. 6 and Fig. 7.","section":"Section 5, Figures"},{"comment":"References [37] and [38] appear in the bibliography but are not cited anywhere in the text.","section":"References"},{"comment":"The statement that 'the computational time for the quantum annealer is 15 minutes to solve the problem' is ambiguous; specify whether this is QPU time, wall-clock time, or total hybrid runtime, and include standard deviations if available.","section":"Section 5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is not ready for publication in its current form. The missing QUBO equations and the contradictory table/text are serious but fixable. The larger concern is the absence of any reproducibility package (code, penalty weights, D-Wave parameters) and the lack of a classical baseline or statistical analysis, which makes the reported performance claims difficult to evaluate. I would be willing to reconsider a substantially revised version that addresses these points, but the current version does not meet the standards of the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The H3S pipeline is the one genuine increment: after FCM clustering, it solves a CVRP on cluster centroids and then expands each truck's assigned clusters into TSPs. That intermediate compression step is a reasonable extension of Feld et al. and CC-CVRP, and it deserves credit as a new combination. The authors also cite the right prior work and are transparent that the pattern in Table 3 is preliminary.\n\nUnfortunately, the load-bearing parts don't hold up. The QUBO Hamiltonians in Section 4.4.2 are garbled; the objective and several constraint equations are incomplete, so I cannot check whether the annealer's output is decoded into feasible CVRP tours. That matters because the paper itself reports an infeasible H2S run for A-n33-k6, an instance that never appears in Table 2 while the text compares against it. The reported costs of 855, 873, 874, 896, 766 are therefore unverified as valid CVRP solutions, and the central 'promising results' claim rests on that.\n\nOn top of that, the experimental evidence is thin: five instances at 32-37 customers, a single run per instance, no error bars, no classical baseline, and a reported 15-minute annealer time that sits oddly with the 'real-time' framing. The manual classification in Table 3 is honestly labeled as needing further testing, so I don't hold that against them.\n\nWho is this for? Someone surveying hybrid QA routing might note the H3S construction as an idea to test seriously. But as a demonstration, it doesn't provide enough to verify feasibility or compare performance.\n\nRecommendation: I would not send this to peer review as is. It deserves a desk reject with an invitation to resubmit once the QUBO formulations are complete, code and data are available, and baselines and multiple runs are reported. If the authors supply that, the H3S idea is worth a careful look.","headline":"The H3S pipeline is a legitimate small extension of Feld et al., but the paper never fully specifies the QUBO models, so the reported route costs cannot be verified as feasible CVRP solutions.","tokens_in":13253,"tokens_out":2783,"would_cite":false,"duration_ms":23507,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["90C27","90C59","81P68"],"pacs":[],"model":"deepseek-v4-flash","headline":"Splitting CVRP into fuzzy clustering plus quantum-annealed routing yields five-instance tours within 17% of best-known costs in about 15 minutes each.","keywords":["quantum annealing","capacitated vehicle routing problem","QUBO","fuzzy C-means clustering","hybrid classical-quantum algorithm","traveling salesman problem","route optimization","logistics optimization"],"falsifier":"Run one of the reported instances (for example A-n32-k5) with the H3S QUBO on a classical simulator, decode the lowest-energy samples into routes, and check every route against the capacity and single-visit constraints; if any decoded low-energy sample violates a constraint, then the Table 2 route costs are not valid CVRP solutions and the central claim collapses.","tokens_in":12187,"feed_emoji":"🚚","tokens_out":6528,"duration_ms":55293,"temperature":0.7,"pith_summary":"This paper tries to establish that a hybrid classical-quantum pipeline can produce usable approximate solutions to the Capacitated Vehicle Routing Problem (CVRP) in about fifteen minutes per instance. The authors split the problem into a capacitated clustering phase, run on a classical computer with fuzzy C-means, and a routing phase whose TSP/CVRP subproblems are encoded as QUBO Hamiltonians and minimized on a quantum annealer. They propose two variants: H2S, which routes each cluster directly as a TSP, and H3S, which adds an intermediate CVRP layer over cluster centroids before expanding back to customers. On five benchmark instances with 32 to 37 customers, both return tours with optimality gaps of roughly 5 to 17 percent above known best solutions, and one further H2S instance produced no feasible solution. If the QUBO encoding is correct, this supports the idea that quantum annealing can assist classical solvers for small real-time routing problems.","feed_headline":"Hybrid quantum routing lands within 17% of best-known truck routes","feed_subtitle":"Fuzzy clustering plus quantum annealing solves five small CVRP instances in about 15 minutes each.","key_machinery":"The load-bearing object is the QUBO Hamiltonian built from binary edge variables $x_{ij}^t$ indicating that truck $t$ traverses edge $(i,j)$. The objective is the sum of edge costs, and the constraints — each customer visited once, each truck leaves the depot once, route continuity, capacity, and subtour elimination via a step variable — are folded in as Lagrange-weighted penalties, with the same model reused for TSP by dropping the capacity constraint. Around this, the classical side is Fuzzy C-Means clustering with depot copies and an iterative preference-and-capacity assignment module; H3S adds a compressed CVRP layer on cluster centroids before the final TSP expansion.","core_discovery":"The paper's central claim is that the H2S and H3S strategies produce promising results in both solution time and solution cost. Concretely, after fuzzy C-means clustering with depot copies and capacity-aware assignment, each cluster is routed either as a TSP (H2S) or through a compressed CVRP over centroids followed by cluster expansion and TSP (H3S). The QUBO Hamiltonians are minimized by quantum annealing, and Table 2 reports total route costs of 695 to 938 against best-known values of 661 to 799 for five instances, corresponding to gaps around 5 to 17 percent, with the annealer taking roughly 15 minutes per instance. The paper also makes a preliminary observation that H3S tends to do better when the depot lies at the corner of the instance and H2S when the depot is central.","pith_inferences":["The real bottleneck may be the clustering assignment rather than the annealer: any error in assigning a node to a wrong cluster is baked into the final route cost, so better capacity-aware clustering could shrink the gap more than a better annealer.","A natural testable extension is to replace the quantum TSP solver with a classical exact TSP solver on the same clusters; the gap between those results and the quantum ones would isolate the annealer's contribution to the optimality gap.","The five-instance depot-location pattern is underpowered, but it suggests a meta-heuristic: use instance geometry to pick the decomposition strategy adaptively."],"forward_implications":["If the QUBO encoding is sound, the two-phase decomposition lets a current quantum annealer return feasible CVRP tours for instances with 32 to 37 customers in about 15 minutes, a scale beyond what direct annealer TSP formulations handled in earlier work.","The H3S compressed-cluster step gives a way to assign multiple clusters to one truck, which matters when truck capacity is large relative to cluster demand.","The reported depot-location pattern, if it holds, gives a cheap rule for choosing between H2S and H3S: corner depots favor H3S, central depots favor H2S.","The approach does not claim to beat classical solvers; it claims a practical quality-time tradeoff for near-real-time logistics."],"supporting_citations":[{"why":"Supplies the QUBO formulation conventions used to turn CVRP/TSP costs and constraints into a binary quadratic Hamiltonian.","marker":"[15]"},{"why":"The hybrid clustering-plus-routing method this paper adapts, solving routing with a quantum annealer after classical clustering.","marker":"[34]"},{"why":"The constrained clustering CVRP decomposition whose compressed-node idea directly inspired the H3S intermediate layer.","marker":"[35]"},{"why":"Establishes the baseline that direct TSP QUBO formulations on a quantum annealer handle at most about eight customers, motivating the hybrid decomposition.","marker":"[30]"},{"why":"Provides a CVRP QUBO formulation with timeline variables used as a reference for the paper's capacity and subtour constraints.","marker":"[33]"},{"why":"Source of the benchmark instances and best-known solution values used in Table 2.","marker":"[13]"}],"fun_headline_variants":["Quantum annealing hybrid cuts route costs to within 17% of best","Hybrid quantum routing: near-best routes in minutes","Quantum annealing speeds route optimization with 17% gap","Quantum hybrid approach finds routes within 17% of best","17% gap: quantum annealing tackles truck routing"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central validity assumption is that the QUBO Hamiltonians used in Section 4.4.2 encode the CVRP constraints correctly, so that any low-energy bitstring the annealer returns decodes to a feasible set of truck routes.","fun_headline_variants_meta":{"raw":{"variants":["Quantum annealing hybrid cuts route costs to within 17% of best","Hybrid quantum routing: near-best routes in minutes","Quantum annealing speeds route optimization with 17% gap","Quantum hybrid approach finds routes within 17% of best","17% gap: quantum annealing tackles truck routing"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001009,"raw_usage":{"total_tokens":4222,"prompt_tokens":863,"completion_tokens":3359,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":479,"completion_tokens_details":{"reasoning_tokens":3280}},"tokens_in":479,"tokens_out":3359,"duration_ms":21286,"temperature":1.0,"reasoning_tokens":3280,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:26:21.039637+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run one of the reported instances (for example A-n32-k5) with the H3S QUBO on a classical simulator, decode the lowest-energy samples into routes, and check every route against the capacity and single-visit constraints; if any decoded low-energy sample violates a constraint, then the Table 2 route costs are not valid CVRP solutions and the central claim collapses.","supporting_citations":[{"cited_title":"A hybrid solution method for the capacitated vehicle routing problem using a quantum annealer","cited_arxiv_id":null,"evidence_quote":"The hybrid clustering-plus-routing method this paper adapts, solving routing with a quantum annealer after classical clustering."},{"cited_title":"Constrained clustering for the capacitated vehicle routing problem (CC-CVRP)","cited_arxiv_id":null,"evidence_quote":"The constrained clustering CVRP decomposition whose compressed-node idea directly inspired the H3S intermediate layer."},{"cited_title":"Solving the traveling salesman problem on the D-Wave quantum computer","cited_arxiv_id":null,"evidence_quote":"Establishes the baseline that direct TSP QUBO formulations on a quantum annealer handle at most about eight customers, motivating the hybrid decomposition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides a CVRP QUBO formulation with timeline variables used as a reference for the paper's capacity and subtour constraints."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Source of the benchmark instances and best-known solution values used in Table 2."}],"review_version":1}