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    "summary": "Kuhn combines bipartite matching and an adequate-budget dual certificate into a finite method for maximum-score one-to-one assignment with positive integer ratings, supplying explicit transfer and budget-improvement routines.\n\nChoosing the best worker for each job separately can assign the same person twice and miss the best overall arrangement. Kuhn instead treats the whole assignment together. A budget attached to every person and job gives an upper bound on any complete assignment; an assignment that exactly meets this budget is therefore optimal. His method alternates between improving a matching along transfers and changing the budget to create new eligible pairs. It turns ideas from graph theory and optimization duality into an explicit solution procedure, rather than checking every possible assignment. The original paper includes proofs, routines and a worked example; later implementations and complexity improvements remain separate work.",
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    "area": "Weighted bipartite assignment",
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        "description": "The method solves the finite assignment problem under the stated cost and matching model. Dual potentials and admissible augmentations are essential. Original algorithmic presentation and later polynomial implementations have different complexity accounts. This is not a solver for arbitrary generalized scheduling or transportation constraints; earlier Kőnig/Egerváry work retains credit.",
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      "statement": "# Kuhn: solve the whole assignment, with a certificate\n\nKuhn combines bipartite matching and an adequate-budget dual certificate into a finite method for maximum-score one-to-one assignment with positive integer ratings, supplying explicit transfer and budget-improvement routines.\n\nChoosing the best worker for each job separately can assign the same person twice and miss the best overall arrangement. Kuhn instead treats the whole assignment together. A budget attached to every person and job gives an upper bound on any complete assignment; an assignment that exactly meets this budget is therefore optimal. His method alternates between improving a matching along transfers and changing the budget to create new eligible pairs. It turns ideas from graph theory and optimization duality into an explicit solution procedure, rather than checking every possible assignment. The original paper includes proofs, routines and a worked example; later implementations and complexity improvements remain separate work.\n\n## Scientific scope\n\nThe method solves the finite assignment problem under the stated cost and matching model. Dual potentials and admissible augmentations are essential. Original algorithmic presentation and later polynomial implementations have different complexity accounts. This is not a solver for arbitrary generalized scheduling or transportation constraints; earlier Kőnig/Egerváry work retains credit.\n\n## People and systems\n\nHarold W. Kuhn\n\n\n\n## Original sources\n\n- [Kuhn, The Hungarian Method for the Assignment Problem (1955), original83-97 facsimile in2010 volume](https://sites.math.washington.edu/~thomas/teaching/m514_web/50YearsIP.pdf)\n- [Harold W.Kuhn: authored introduction to the Hungarian-method reprint (2010)](https://sites.math.washington.edu/~thomas/teaching/m514_web/50YearsIP.pdf)\n",
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