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    "title": "Munkres: make optimal assignment a bounded procedure",
    "summary": "Munkres gives a constructive starred/primed-zero variant of Kuhn's assignment algorithm, proves finite termination without requiring integer costs and provides an explicit polynomial bound in the paper's line-operation model, then states a transportation generalization.\n\nSuppose every possible worker-job pair has a cost. Trying all complete assignments quickly becomes impractical, because their number grows factorially. Munkres explains how to build an optimal assignment by marking zero-cost choices in an adjusted cost matrix, following alternating chains and changing the matrix only when needed. The adjustments preserve which complete assignment is best. His analysis shows that progress does not depend on costs being integers, and gives a bound expressed in specified elementary operations. The same idea is adapted to moving integer quantities between sources and destinations. This is a refinement of an existing optimization method, with its own procedure and analysis, rather than a new invention of the entire assignment problem or all later Hungarian-algorithm improvements.",
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        "description": "archival cover PDF 1 excluded from scientific evidence. Finite square linear assignment with independent one-per-row/column entries; initially integer costs, later integer assumption explicitly removed by bounded progress of covers/starred zeros. Polynomial bound counts whole-line scans/updates as specified operations, not unit-cost scalar instructions or a verified modern O(n^3) implementation. The exact operation-count coefficients and the complete cost-model derivation have not been independently verified here. Transportation section assumes nonnegative integer costs and positive integer supplies/demands with equal totals, provides an integral-quota procedure and omits detailed theoretical remarks. No complete transportation-proof reconstruction or extension to arbitrary real supplies, quadratic assignment, all network flows, bitcomplexity or empirical speedup is imported. Later algorithmic improvements and library implementations are separate; neither all optimal assignments nor unique optimality is claimed.",
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      "statement": "# Munkres: make optimal assignment a bounded procedure\n\nMunkres gives a constructive starred/primed-zero variant of Kuhn's assignment algorithm, proves finite termination without requiring integer costs and provides an explicit polynomial bound in the paper's line-operation model, then states a transportation generalization.\n\nSuppose every possible worker-job pair has a cost. Trying all complete assignments quickly becomes impractical, because their number grows factorially. Munkres explains how to build an optimal assignment by marking zero-cost choices in an adjusted cost matrix, following alternating chains and changing the matrix only when needed. The adjustments preserve which complete assignment is best. His analysis shows that progress does not depend on costs being integers, and gives a bound expressed in specified elementary operations. The same idea is adapted to moving integer quantities between sources and destinations. This is a refinement of an existing optimization method, with its own procedure and analysis, rather than a new invention of the entire assignment problem or all later Hungarian-algorithm improvements.\n\n## Scientific scope\n\narchival cover PDF 1 excluded from scientific evidence. Finite square linear assignment with independent one-per-row/column entries; initially integer costs, later integer assumption explicitly removed by bounded progress of covers/starred zeros. Polynomial bound counts whole-line scans/updates as specified operations, not unit-cost scalar instructions or a verified modern O(n^3) implementation. The exact operation-count coefficients and the complete cost-model derivation have not been independently verified here. Transportation section assumes nonnegative integer costs and positive integer supplies/demands with equal totals, provides an integral-quota procedure and omits detailed theoretical remarks. No complete transportation-proof reconstruction or extension to arbitrary real supplies, quadratic assignment, all network flows, bitcomplexity or empirical speedup is imported. Later algorithmic improvements and library implementations are separate; neither all optimal assignments nor unique optimality is claimed.\n\n## People and systems\n\nJames Munkres\n\n\n\n## Original sources\n\n- [James Munkres, Algorithms for the Assignment and Transportation Problems (1957),32-38](https://www.math.ucdavis.edu/~saito/data/emd/munkres.pdf)\n",
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