{ "schemaVersion": "1.6", "slug": "integer-calendar-replenishment", "title": "Integer calendars for fixed-turnover replenishment: a 1/ln 2 guarantee on trees, the price of short cycles and the price of constant gaps", "shortTitle": "From factor 2 to 1/ln 2 for recurring deliveries", "url": "https://evidencepress.org/releases/integer-calendar-replenishment/", "oneLine": "Answering Bosman et al.'s 2022 question: rounding each client's clock events, not its period, to whole days gives delivery calendars that never miss a deadline and cost at most 1/ln 2 ≈ 1.44 times the optimum on trees, improving the factor two; constant-gap calendars can reach only about 1.60.", "abstract": "A depot must revisit every client at least once within that client's turnover time while keeping the long-run average daily routing cost low. Bosman et al. (2022) gave a factor-two approximation on tree networks and asked whether it could be improved. This candidate shows that it can. Every client gets a real-valued clock from one shared power-of-two ladder, the ladder's phase is chosen by a logarithmic averaging argument and an exact breakpoint sweep, and each clock's event times are rounded up to whole days. The calendar meets every deadline and costs at most 1/ln 2 ≈ 1.4427 times the optimum, for any monotone submodular daily cost, of which tree routing is one case; it is computed deterministically in O(n log n) time. It repeats within the longest turnover time, and for such calendars 1/ln 2 is the exact worst-case price; the analysis of the algorithm is tight on trees. Individual gaps may alternate between two consecutive integers. If every client must keep a constant gap, with the gaps dividing one another (the form of Bosman et al.'s output), the best possible worst-case factor lies between 1.6001 and 1.6014, with an explicit four-ladder mixture giving 48/29. The paper also solves the nested two-edge problem exactly, gives divisibility and least-common-multiple criteria for when the frequency lower bound is attained, profiles the cost of longer cycles on the worst-case family, and certifies instances where the optimum exceeds that bound by a factor 1.204. The rounding tools and the constant are classical. AI-assisted, unrefereed candidate; a 2025 master's thesis with a closely matching title could not be obtained, so priority is not claimed; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-10-31", "dateModified": "2026-10-31", "version": "1.0.0-candidate", "doi": "10.5281/zenodo.23845965", "doiUrl": "https://doi.org/10.5281/zenodo.23845965", "conceptDoi": "10.5281/zenodo.23845964", "pdfUrl": "https://github.com/ipitchford/integer-calendar-replenishment/releases/download/v1.0.0-candidate/integer-calendar-replenishment-1.0.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23845965", "repoUrl": "https://github.com/ipitchford/integer-calendar-replenishment", "releaseUrl": "https://github.com/ipitchford/integer-calendar-replenishment/releases/tag/v1.0.0-candidate", "markdownUrl": "https://evidencepress.org/releases/integer-calendar-replenishment/index.md", "bibtexUrl": "https://evidencepress.org/releases/integer-calendar-replenishment/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/integer-calendar-replenishment.mp3", "imageUrl": "https://evidencepress.org/assets/og/integer-calendar-replenishment.png?v=d68dda0147", "coverArtUrl": "https://evidencepress.org/assets/art/integer-calendar-replenishment.svg?v=7f8ad7b8f4", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/integer-calendar-replenishment.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/integer-calendar-replenishment.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted mathematical candidate. Every theorem has a written proof; finite bounds and optima are certified in exact arithmetic. Internal AI-assisted review (a five-role first round, an outside-model adversarial review, a clean replay, and a re-review and two proof audits of the revision). The rounding tools, phase sweep and constant are classical and credited. Priority is not claimed: a 2025 DTU MSc thesis with a closely matching title could not be obtained. External specialist review, independent reproduction and formal verification have not been established." }, "assurance": [ { "dimension": "availability", "label": "Availability and archiving", "question": "Is the evidence package publicly retrievable from an archive under a persistent identifier?", "state": "passed", "evidenceUrl": "https://zenodo.org/records/23845965", "note": "The versioned research package and manuscript are publicly archived on GitHub and Zenodo. Provider asset parity was checked against frozen local files; availability does not establish correctness or peer review." }, { "dimension": "internalReplay", "label": "Internal replay", "question": "Does the producer’s own pipeline reproduce the stated result from the archived package?", "state": "passed", "note": "From a fresh extraction of the frozen archive every verifier passed on the shipped certificates in normal and optimised Python, every computation was regenerated into a separate directory with identical optima and counts, and 13 semantic negative controls were rejected. GitHub CI repeats the exact checks on Linux." }, { "dimension": "independentRerun", "label": "Independent rerun", "question": "Has someone else run the supplied implementation and obtained the stated result?", "state": "not-assessed", "note": "No unaffiliated rerun has been established." }, { "dimension": "independentReimplementation", "label": "Independent reimplementation", "question": "Has someone else reached the result from an independent implementation?", "state": "partial", "note": "Separately written calendar checkers, finite-state optimum certificates and two audit re-implementations of the constant-gap bounds agree; this is producer-side implementation diversity, not unaffiliated reimplementation." }, { "dimension": "formalVerification", "label": "Formal verification", "question": "Is a formalised statement machine-checked, and over which trusted base?", "state": "not-assessed", "note": "No proof-assistant formalisation; the theorems are proved in prose, with exact computer certificates for finite bounds and optima." }, { "dimension": "specialistReview", "label": "Specialist review", "question": "Has a domain specialist assessed the argument?", "state": "not-assessed", "note": "AI-assisted internal and adversarial review is not external specialist review." }, { "dimension": "editorialPeerReview", "label": "Editorial peer review", "question": "Has a journal or venue run peer review to a decision?", "state": "not-assessed", "note": "No external editorial peer review." }, { "dimension": "dataEnvironmentReproducibility", "label": "Data and environment reproducibility", "question": "Are data and computational environment pinned well enough to rebuild?", "state": "partial", "note": "The calendar code and every verifier need only the Python standard library; certificate generation pins NumPy 2.3.5 and SciPy 1.17.0 (requirements.txt), not a locked container." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted mathematical research", "AI-assisted internal review, outside-model adversarial review, re-review and proof audits", "Claude Code: publication packaging and release" ], "humanRole": "Publication authority only; no human research contribution inferred.", "reviewStatus": "Internal AI-assisted and adversarial review, not external peer review." }, "attributionBoundary": null, "problem": null, "orderedProductAverage": null, "corrections": [], "publicCorrections": [], "publicCorrectionIndexes": null, "pageStructureVersion": "2.0", "pageStructureVariant": "canonical-theorem", "pageStructureWaivers": [], "recordMaturity": "current", "metadataProvenance": "Prepared from the final manuscript (31 October 2026, 26 pages), its exact verifiers and the frozen release archive. The versioned GitHub and Zenodo research archives were checked against the frozen release assets on 31 October 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "inventory routing", "fixed turnover times", "replenishment", "approximation algorithms", "submodular costs", "harmonic schedules", "pinwheel scheduling", "open problem" ], "keyResults": [ "Integer calendars: rounding the event times of logarithmically phased power-of-two clocks to whole days meets every deadline and costs at most W₁ + (L_f − W₁)/ln 2 ≤ OPT/ln 2 ≈ 1.4427·OPT for any monotone submodular daily cost (Theorem 1); on trees this improves the factor 2 of Bosman et al. (Corollary 1).", "Short cycles: the calendar repeats within the longest turnover τ_max, and sup OPT_short/OPT = 1/ln 2 exactly for calendars with that property; the analysis is tight on trees (Theorem 7, eq. (29), Corollary 2).", "Constant gaps: if every client keeps a constant gap and the gaps divide one another, the best worst-case factor ρ_H satisfies 1.60010 ≤ ρ_H ≤ 1.60139; mixtures of two and four integer ladders give 12/7 and 48/29 (Theorem 9).", "Synchronisation: exact optimum of two nested edges; the frequency bound is attained on a tree iff the edge turnovers divide along every path, and for coverage costs iff lcm{q_A : i ∈ A} ≤ τ_i (Theorems 2, 3, 5).", "The frequency lower bound can be loose: certified broom instances have OPT/L_T = 3780/3139 ≈ 1.204.", "Worked example: deadlines 2 and 3 on a path; the power-of-two calendar costs 3 per day, the integer calendar 8/3, which is optimal." ], "reviews": [], "evidencePackage": "Written proofs of Theorems 1–9, Lemmas 1–4 and Corollaries 1–2; 195 exactly certified optimal schedules for small trees and brooms (attaining cycles and integer potentials) and 100 for coverage costs; exact rational certificates for the constant-gap bounds, including an enumeration of all 15,374 divisor chains up to 511; standard-library verifiers that do not rely on assert, replayed from the frozen archive in normal and optimised Python; 13 semantic negative controls; GitHub CI.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Bosman, van Ee, Jiao, Marchetti-Spaccamela, Ravi and Stougie (2022), Approximation algorithms for replenishment problems with fixed turnover times. Algorithmica 84(9), 2597–2621 (the model, the factor-two tree algorithm and the open question answered here).", "url": "https://doi.org/10.1007/s00453-022-00974-4", "doi": "10.1007/s00453-022-00974-4" }, { "citation": "Bosman, van Ee, Jiao, Marchetti-Spaccamela, Ravi and Stougie (2018), Approximation algorithms for replenishment problems with fixed turnover times. LATIN 2018, Lecture Notes in Computer Science, 217–230 (conference version).", "url": "https://doi.org/10.1007/978-3-319-77404-6_17", "doi": "10.1007/978-3-319-77404-6_17" }, { "citation": "Roundy (1989), Rounding off to powers of two in continuous relaxations of capacitated lot sizing problems. Management Science 35(12), 1433–1442 (classical source of the 1/ln 2 rounding factor).", "url": "https://doi.org/10.1287/mnsc.35.12.1433", "doi": "10.1287/mnsc.35.12.1433" }, { "citation": "Teo and Bertsimas (2001), Multistage lot sizing problems via randomized rounding. Operations Research 49(4), 599–608 (logarithmic random shifts and the greedy submodular representation).", "url": "https://doi.org/10.1287/opre.49.4.599.11222", "doi": "10.1287/opre.49.4.599.11222" }, { "citation": "Federgruen and Zheng (1993), Optimal power-of-two replenishment strategies in capacitated general production/distribution networks. Management Science 39(6), 710–727 (power-of-two policies with bounds on reorder intervals).", "url": "https://doi.org/10.1287/mnsc.39.6.710", "doi": "10.1287/mnsc.39.6.710" }, { "citation": "Chan and Chin (1993), Schedulers for larger classes of pinwheel instances. Algorithmica 9(5), 425–462 (single-integer reduction; the octave witness family in pinwheel scheduling).", "url": "https://doi.org/10.1007/BF01187034", "doi": "10.1007/BF01187034" }, { "citation": "Kawamura (2024), Proof of the density threshold conjecture for pinwheel scheduling. Proceedings of the 56th Annual ACM Symposium on Theory of Computing, 1816–1819.", "url": "https://doi.org/10.1145/3618260.3649757", "doi": "10.1145/3618260.3649757" }, { "citation": "Fujiwara, Miyagi and Ouchi (2026), Pinwheel scheduling with real periods. Discrete Mathematics & Theoretical Computer Science 28(4) (real periods realised on integer days).", "url": "https://doi.org/10.46298/dmtcs.17657", "doi": "10.46298/dmtcs.17657" }, { "citation": "Díaz-Ramírez, Mejía-Alvarez and Leyva-del-Foyo (2013), Comprehensive comparison of schedulability tests for uniprocessor rate-monotonic scheduling. Journal of Applied Research and Technology 11(3), 408–436 (reproduces the Han–Tyan harmonic sweep used for clock selection).", "url": "https://doi.org/10.1016/S1665-6423(13)71551-7", "doi": "10.1016/S1665-6423(13)71551-7" }, { "citation": "Liu and Layland (1973), Scheduling algorithms for multiprogramming in a hard-real-time environment. Journal of the ACM 20(1), 46–61 (the ln 2 utilisation bound).", "url": "https://doi.org/10.1145/321738.321743", "doi": "10.1145/321738.321743" }, { "citation": "Bosman and Olver (2020), Improved approximation algorithms for inventory problems. Integer Programming and Combinatorial Optimization, Lecture Notes in Computer Science, 91–103 (general-metric state of the art).", "url": "https://doi.org/10.1007/978-3-030-45771-6_8", "doi": "10.1007/978-3-030-45771-6_8" }, { "citation": "Bosman (2019), Relax, round, reformulate: near-optimal algorithms for planning problems in network design and scheduling. Doctoral thesis, Vrije Universiteit Amsterdam (Chapter 3 restates the factor two and the open question).", "url": "https://ir.cwi.nl/pub/29320", "doi": null } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:integer-calendar-replenishment", "attemptIds": [ "ep-attempt:integer-calendar-replenishment-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "certificate-first", "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "bound", "description": "Answers the open question of Bosman et al. (2022, Section 5): the factor-two approximation for min-avg replenishment with fixed turnover times on trees improves to 1/ln 2 with integer-day calendars, exact for calendars repeating within the longest turnover; constant-gap harmonic calendars have best factor between 1.6001 and 1.6014.", "scope": "Deterministic integer turnover times, long-run average cost, monotone submodular daily costs (trees). No capacity, holding cost, stochastic demand or peak-day objective; general metrics, hardness and general constant-gap policies open." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "The feasibility model and optimum are exactly those of Bosman et al. (2022, Section 1); the comparison uses their tree lower bound (Lemma 2). Their outputs are constant-gap; the 1/ln 2 calendars are not, which is stated, and Theorem 9 treats the constant-gap class separately.", "remainingRisks": [ "An unobtained 2025 MSc thesis with a closely matching title may contain an equivalent result.", "Distance-constrained scheduling papers (Han and Lin) could not be inspected in full and may contain the integer-slot rounding." ] }, "humanJudgmentGates": [ "Publication is authorised; scientific endorsement is not inferred.", "External specialist review, independent reproduction and the thesis priority check remain open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "specialistReview", "independentRerun", "formalVerification", "priorityAssessment" ], "nextAction": "Obtain the 2025 DTU thesis or an account of its results, and specialist review of Theorems 1 and 9.", "claimCeiling": "Proved 1/ln 2 integer-calendar guarantee and exact short-cycle price; certified constant-gap bracket; priority not claimed. 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