{ "schemaVersion": "1.6", "slug": "vector-matching-linear-discrepancy", "title": "Linear discrepancy for vector-labelled perfect matchings", "shortTitle": "Fairer pairings: matching error 17k instead of O(k²)", "url": "https://evidencepress.org/releases/vector-matching-linear-discrepancy/", "oneLine": "For edges of a complete bipartite graph labelled by vectors (for example k colours), some perfect matching lands within 17k of any fractional target in any k-dimensional norm, improving the previous O(k²); order k is necessary for arbitrary targets, and the key step cuts all leftover fractional paths at once.", "abstract": "Colour the edges of the complete bipartite graph K_{n,n} with k colours. How close can a perfect matching come to using every colour in its expected proportion? Hogan, Scott and Tsarev (2026) proved that the total imbalance can be kept to O(k²), independently of n; earlier, Hollom gave a bound of order 4^{k²}. This candidate proves a linear bound in a more general setting: if every edge carries a label in the unit ball of a k-dimensional real normed space and x is any fractional perfect matching, some perfect matching M has ‖h(M) − Σ x_e h(e)‖ ≤ 17k. For arbitrary fractional targets the order k is best possible in the worst case over norms (an ℓ₁ example forces k/2); for the uniform target of the colour-balancing problem the known lower bound is √(k/2), so the square-root conjecture stays open. The key tool is a splitting lemma in the Hobby–Rice and necklace-splitting tradition: d signed measures on an interval cut into q pieces, with a different fraction on each piece, are matched by one set with at most q − 1 + 2d switches, instead of about 2qd when the pieces are split separately. Consequences include linear-in-k bounds for colour-balanced perfect matchings of complete graphs, spanning forests, Hamilton cycles and complete multipartite hypergraph matchings, and an existence version of a school-assignment rounding (FORC 2025) with total capacity violation at most 8g − 1 instead of O(g²) for g groups. The proofs are existential; no efficient algorithm is claimed. AI-assisted, unrefereed candidate; no external peer review, independent reproduction or formal verification is claimed.", "datePublished": "2026-11-03", "dateModified": "2026-11-03", "version": "0.1.0-candidate", "doi": "10.5281/zenodo.23932844", "doiUrl": "https://doi.org/10.5281/zenodo.23932844", "conceptDoi": "10.5281/zenodo.23932843", "pdfUrl": "https://github.com/ipitchford/vector-matching-linear-discrepancy/releases/download/v0.1.0-candidate/vector-matching-linear-discrepancy-0.1.0-candidate.pdf", "altPdfUrl": null, "altPdfLabel": null, "zenodoUrl": "https://zenodo.org/records/23932844", "repoUrl": "https://github.com/ipitchford/vector-matching-linear-discrepancy", "releaseUrl": "https://github.com/ipitchford/vector-matching-linear-discrepancy/releases/tag/v0.1.0-candidate", "markdownUrl": "https://evidencepress.org/releases/vector-matching-linear-discrepancy/index.md", "bibtexUrl": "https://evidencepress.org/releases/vector-matching-linear-discrepancy/cite.bib", "audioUrl": "https://evidencepress.org/assets/audio/vector-matching-linear-discrepancy.mp3", "imageUrl": "https://evidencepress.org/assets/og/vector-matching-linear-discrepancy.png?v=11737eaecb", "coverArtUrl": "https://evidencepress.org/assets/art/vector-matching-linear-discrepancy.svg?v=f94daab6f7", "media": [ { "type": "audio", "url": "https://evidencepress.org/assets/audio/vector-matching-linear-discrepancy.mp3", "name": "Plain-English audio briefing", "description": "Communication aid; not additional research evidence.", "transcriptUrl": "https://evidencepress.org/assets/audio/vector-matching-linear-discrepancy.txt" } ], "authors": [ "Anonymous" ], "license": "CC0-1.0", "status": "unrefereed-candidate", "verification": { "peerReviewed": false, "independentlyReproduced": false, "formallyVerified": false, "internallyReplayed": true, "detail": "Unrefereed AI-assisted combinatorics candidate. Every result has a written proof; the checker verifies finite bookkeeping, not the Hobby–Rice step, the compactness argument or statements for all n and k. Existence proofs only; no algorithm is claimed. Internal AI-assisted review in five roles, a separate-model adversarial review and audit, a line-by-line proof audit, a clean replay, one revision round and a re-review. 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/23932844", "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": "check_lemmas.py was rerun with Python -I -B, with and without -O, from a fresh copy of the approved package; both outputs matched the retained verification_results.json apart from timing and platform fields, and the mutation self-test caught both corrupted cases. make_figure.py reproduced the PNG figure byte for byte. The frozen archive was bound to that replay and the checker rerun from it; GitHub CI reruns 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": "not-assessed", "note": "No independent reimplementation has been established." }, { "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." }, { "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 checker needs only the Python standard library; the figure pins Matplotlib 3.10.8 in requirements.txt; not a locked container." } ], "provenance": { "aiGenerated": true, "aiAssisted": true, "generatedBy": [ "AI-assisted combinatorics and computation (ChatGPT for the original draft; Claude-based agents for the revision)", "AI-assisted internal review in five roles (Claude Code subagents); adversarial review and audit of the new proofs by a separate model (OpenAI Codex); line-by-line proof audit; clean replay; re-review", "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 (revision of 3 November 2026 with DOI insertion, 16 pages), its verification script and the frozen release archive. The versioned GitHub and Zenodo research archives were checked against the frozen release assets on 3 November 2026.", "grandfatheredAtSchemaVersion": null, "keywords": [ "perfect matchings", "discrepancy", "colour-balanced matchings", "fractional perfect matching", "rounding", "necklace splitting", "Hobby–Rice theorem", "group fairness", "school assignment" ], "keyResults": [ "Linear bound (Theorem 1, proof): for labels in the unit ball of any k-dimensional real normed space on the edges of K_{n,n} and any fractional perfect matching x, some perfect matching M has ‖h(M) − Σ x_e h(e)‖ ≤ 17k; for the uniform target in ℓ₁^k this improves O(k²) (Hogan, Scott and Tsarev) to 17k.", "Sharpness for arbitrary targets (Proposition 2, Remark 13, proof and exact checks): an ℓ₁^k example forces discrepancy exactly k/2; the same example gives √k/2 in ℓ₂^k and 1/2 in ℓ∞^k, and a Hadamard example gives √k/2 in ℓ∞^k for k a power of two. For the uniform target the order lies between √(k/2) and 17k.", "Splitting lemma (Lemma 5, proof): d finite atomless signed measures on an interval cut into q pieces, with fraction α_j wanted on piece j, are matched by one finite union of intervals with at most q − 1 + 2d switches, instead of about 2qd when the pieces are split separately.", "Consequences (Corollaries 7, 10, 11, Propositions 8–9, proof): colour-balanced perfect matchings of K_{2kt} within 17k + 2√k; spanning forests O(Δk); Hamilton cycles O(k); complete s-partite s-uniform hypergraph matchings 17(s − 1)k.", "School assignment (Corollary 12, proof): in the setting of Santhini, Munagala, Nasre and Sankar (FORC 2025) there is an integral assignment meeting every group's optimal utility with total capacity violation at most 8g − 1, instead of O(g²); existence only." ], "reviews": [], "evidencePackage": "Written proofs of every lemma, theorem, proposition and corollary; check_lemmas.py (Python standard library only) with 2,374,625 exact integer and rational checks of the finite bookkeeping: the 64 local configurations of Lemma 4, 87,380 cut patterns of Lemma 6, 510 cases of the binary recursion of Lemma 5 and its switch budget, the lower-bound example in ℓ₁, ℓ₂ and ℓ∞ for k ≤ 8, the constant arithmetic for k ≤ 40 and 335,922 overload patterns of Corollary 12, each raising on failure so they stay active under Python -O, plus a mutation self-test; clean replay from a fresh copy with and without -O reproducing the retained output, and the figure script reproducing the PNG figure byte for byte.", "openProblems": [], "mathObjects": [], "relatedWorks": [ { "citation": "Hogan, Scott and Tsarev (2026). Colour-balanced subgraphs. arXiv:2604.09449v2. The O(k²) matching bound (Theorem 1.9) improved here to 17k, the uniform-target lower bound √(k/2) and the conjectures left open.", "url": "https://arxiv.org/abs/2604.09449v2", "doi": "10.48550/arXiv.2604.09449" }, { "citation": "Hollom (2025). A Uniform Bound on Almost Colour-Balanced Perfect Matchings in Colour-Balanced Complete Graphs. The Electronic Journal of Combinatorics. The first bound independent of t, 4^{k²}.", "url": "https://doi.org/10.37236/13491", "doi": "10.37236/13491" }, { "citation": "Pardey and Rautenbach (2022). Almost color-balanced perfect matchings in color-balanced complete graphs. Discrete Mathematics. Showed exact balance can fail and conjectured O(k²).", "url": "https://doi.org/10.1016/j.disc.2021.112701", "doi": "10.1016/j.disc.2021.112701" }, { "citation": "Kittipassorn and Sinsap (2023). On the existence of zero-sum perfect matchings of complete graphs. Ars Mathematica Contemporanea. Origin of the colour-balanced matching question.", "url": "https://doi.org/10.26493/1855-3974.2573.90d", "doi": "10.26493/1855-3974.2573.90d" }, { "citation": "Santhini K. A., Munagala, Nasre and Sankar (2025). Group Fairness and Multi-Criteria Optimization in School Assignment. FORC 2025, LIPIcs 329. The school-assignment rounding whose O(g²) capacity violation Corollary 12 lowers to 8g−1 (existence only).", "url": "https://doi.org/10.4230/lipics.forc.2025.20", "doi": "10.4230/lipics.forc.2025.20" }, { "citation": "Hobby and Rice (1965). A Moment Problem in L1 Approximation. Proceedings of the American Mathematical Society. The splitting theorem behind Lemma 3.", "url": "https://doi.org/10.2307/2033900", "doi": "10.2307/2033900" }, { "citation": "Pinkus (1976). A simple proof of the Hobby-Rice theorem. Proceedings of the American Mathematical Society. The form of Lemma 3 used.", "url": "https://doi.org/10.1090/s0002-9939-1976-0425470-0", "doi": "10.1090/s0002-9939-1976-0425470-0" }, { "citation": "Stromquist and Woodall (1985). Sets on which several measures agree. Journal of Mathematical Analysis and Applications. The halving-and-closure template followed by Lemma 5.", "url": "https://doi.org/10.1016/0022-247x(85)90021-6", "doi": "10.1016/0022-247x(85)90021-6" }, { "citation": "Alon (1987). Splitting necklaces. Advances in Mathematics. Necklace splitting with few cuts.", "url": "https://doi.org/10.1016/0001-8708(87)90055-7", "doi": "10.1016/0001-8708(87)90055-7" }, { "citation": "Simmons and Su (2003). Consensus-halving via theorems of Borsuk-Ulam and Tucker. Mathematical Social Sciences. The fair-division form of Lemma 3.", "url": "https://doi.org/10.1016/s0165-4896(02)00087-2", "doi": "10.1016/s0165-4896(02)00087-2" }, { "citation": "Akopyan and Karasev (2013). Cutting the Same Fraction of Several Measures. Discrete & Computational Geometry. Ham-sandwich theorem for charges used in the common-fraction case.", "url": "https://doi.org/10.1007/s00454-012-9450-4", "doi": "10.1007/s00454-012-9450-4" }, { "citation": "Grandoni, Ravi, Singh and Zenklusen (2014). New approaches to multi-objective optimization. Mathematical Programming. Common-proportion splitting of signed measures; quadratic loss in the budget count.", "url": "https://doi.org/10.1007/s10107-013-0703-7", "doi": "10.1007/s10107-013-0703-7" }, { "citation": "Chekuri, Vondrák and Zenklusen (2011). Multi-budgeted Matchings and Matroid Intersection via Dependent Rounding. SODA 2011. Dependent rounding of fractional matchings, a different guarantee.", "url": "https://doi.org/10.1137/1.9781611973082.82", "doi": "10.1137/1.9781611973082.82" }, { "citation": "Lenstra, Shmoys and Tardos (1990). Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programming. The extreme-point counting standard in iterative rounding.", "url": "https://doi.org/10.1007/bf01585745", "doi": "10.1007/bf01585745" } ], "claimAssurance": null, "operatingModel": { "version": "1.0", "workId": "ep-work:vector-matching-linear-discrepancy", "attemptIds": [ "ep-attempt:vector-matching-linear-discrepancy-publication" ], "aims": [ "science" ], "artifactRoles": [ "research-output", "communication" ], "lineageId": null, "accelerationPrimitives": [ "adversarial-controls", "assurance-vector", "agent-readable-research-object" ], "decisionObject": { "type": "bound", "description": "How far the best perfect matching must be from a fractional target when edges carry vector labels: at most 17k in any k-dimensional norm, order k necessary for arbitrary targets, and a joint path-splitting lemma that turns per-path quadratic rounding loss into linear loss.", "scope": "Complete bipartite graphs and the reductions of Hogan, Scott and Tsarev; the FORC 2025 school-assignment setting; existence only." }, "bottleneckTargeted": [ "assurance", "publication" ], "semanticBridge": { "state": "explicit", "description": "Section 1.3 and its comparison table separate the per-path splitting of Hogan, Scott and Tsarev and of the FORC 2025 paper from the joint unequal-fraction splitting of Lemma 5; Section 5.5 quotes the FORC model and theorem it adapts; Section 7 records exactly what the checker covers.", "remainingRisks": [ "The bounded literature check is not exhaustive priority certification; Lemma 5 may be known in necklace-splitting or moment-theory language.", "The constant 17 is not optimised and no algorithm is given." ] }, "humanJudgmentGates": [ "Publication is authorised; scientific endorsement is not inferred.", "External specialist review of Lemma 5 and independent reproduction remain open." ], "parentLinks": [], "assuranceTarget": { "dimensions": [ "specialistReview", "independentRerun", "formalVerification", "priorityAssessment" ], "nextAction": "Have a specialist in necklace splitting check whether Lemma 5 (unequal fractions on q pieces with q − 1 + 2d switches) is known, and a combinatorialist check the deletion and completion counts behind the constant 17.", "claimCeiling": "Proved 17k bound for every fractional target and every k-dimensional norm, order-k sharpness for arbitrary targets, the splitting lemma and its consequences; exact finite checks. 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