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    "title": "Gomory: cutting away a fractional answer without losing integer solutions",
    "summary": "Gomory derives an integer-valid constraint from a fractional row of a linear-programming tableau, excluding the current fractional answer while retaining integer solutions. His 1958 announcement outlines an integer-programming algorithm but defers its full termination rule and proof.\n\nA linear optimizer may recommend half a machine or a fraction of a shipment when the decision must be a whole number. Simply rounding can break other constraints. Gomory instead derives a new constraint from the fractional parts of an existing equation. This cut removes the current fractional answer while preserving every integer answer allowed by that equation. The linear problem can then be optimized again with the new constraint. The note explains why the cut is valid and reports tests on small problems. It also makes a crucial distinction: the special choices that guarantee eventual termination, and their complete proof, are deferred to a longer treatment. The contribution recorded here is the automatic integer-valid cut construction and the algorithm announcement, rather than a complete proof of every cutting strategy or a modern mixed-integer solver.",
    "boundary": "Exact 1958 Bulletin AMS 64(5), 275-278 announcement; Damaged tableau notation and computer-model text are not certified transcriptions. The displayed construction uses nonnegative integer variables, including objective and slack variables; modern mixed-integer cuts are a different target. Page 276 and page 278 explicitly defer the complete termination-enforcing choices and finiteness proof to a longer paper. The supplied cut-validity explanation does not make arbitrary row-selection heuristics convergent. Earlier manually constructed cuts by Dantzig/Fulkerson/Johnson and Markowitz/Manne are acknowledged. The largest reported small test had at most seven variables; no modern solver benchmark or general efficiency claim is established. Row selection is nonunique and the suggested deepest-fraction rule is heuristic. A. Tucker communicated the note and Princeton-IBM supplied institutional support; neither role is coauthorship. Later complete algorithms, mixed-integer variants and branch-and-cut implementations remain separate.",
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        "description": "Exact 1958 Bulletin AMS 64(5), 275-278 announcement; Damaged tableau notation and computer-model text are not certified transcriptions. The displayed construction uses nonnegative integer variables, including objective and slack variables; modern mixed-integer cuts are a different target. Page 276 and page 278 explicitly defer the complete termination-enforcing choices and finiteness proof to a longer paper. The supplied cut-validity explanation does not make arbitrary row-selection heuristics convergent. Earlier manually constructed cuts by Dantzig/Fulkerson/Johnson and Markowitz/Manne are acknowledged. The largest reported small test had at most seven variables; no modern solver benchmark or general efficiency claim is established. Row selection is nonunique and the suggested deepest-fraction rule is heuristic. A. Tucker communicated the note and Princeton-IBM supplied institutional support; neither role is coauthorship. Later complete algorithms, mixed-integer variants and branch-and-cut implementations remain separate.",
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      "statement": "# Gomory: cutting away a fractional answer without losing integer solutions\n\nGomory derives an integer-valid constraint from a fractional row of a linear-programming tableau, excluding the current fractional answer while retaining integer solutions. His 1958 announcement outlines an integer-programming algorithm but defers its full termination rule and proof.\n\nA linear optimizer may recommend half a machine or a fraction of a shipment when the decision must be a whole number. Simply rounding can break other constraints. Gomory instead derives a new constraint from the fractional parts of an existing equation. This cut removes the current fractional answer while preserving every integer answer allowed by that equation. The linear problem can then be optimized again with the new constraint. The note explains why the cut is valid and reports tests on small problems. It also makes a crucial distinction: the special choices that guarantee eventual termination, and their complete proof, are deferred to a longer treatment. The contribution recorded here is the automatic integer-valid cut construction and the algorithm announcement, rather than a complete proof of every cutting strategy or a modern mixed-integer solver.\n\n## Scientific scope\n\nExact 1958 Bulletin AMS 64(5), 275-278 announcement; Damaged tableau notation and computer-model text are not certified transcriptions. The displayed construction uses nonnegative integer variables, including objective and slack variables; modern mixed-integer cuts are a different target. Page 276 and page 278 explicitly defer the complete termination-enforcing choices and finiteness proof to a longer paper. The supplied cut-validity explanation does not make arbitrary row-selection heuristics convergent. Earlier manually constructed cuts by Dantzig/Fulkerson/Johnson and Markowitz/Manne are acknowledged. The largest reported small test had at most seven variables; no modern solver benchmark or general efficiency claim is established. Row selection is nonunique and the suggested deepest-fraction rule is heuristic. A. Tucker communicated the note and Princeton-IBM supplied institutional support; neither role is coauthorship. Later complete algorithms, mixed-integer variants and branch-and-cut implementations remain separate.\n\n## People and systems\n\nRalph E. Gomory\n\n\n\n## Original sources\n\n- [Gomory: cutting away a fractional answer without losing integer solutions; exact original technical paper](https://ralphgomory.org/wp-content/uploads/2023/02/07_outline.pdf)\n- [Publisher-deposited identity for gomory-fractional-integer-cuts-1958](https://api.crossref.org/works/10.1090%2Fs0002-9904-1958-10224-4)\n",
      "summary": "Gomory derives an integer-valid constraint from a fractional row of a linear-programming tableau, excluding the current fractional answer while retaining integer solutions. His 1958 announcement outlines an integer-programming algorithm but defers its full termination rule and proof.\n\nA linear optimizer may recommend half a machine or a fraction of a shipment when the decision must be a whole number. Simply rounding can break other constraints. Gomory instead derives a new constraint from the fractional parts of an existing equation. This cut removes the current fractional answer while preserving every integer answer allowed by that equation. The linear problem can then be optimized again with the new constraint. The note explains why the cut is valid and reports tests on small problems. It also makes a crucial distinction: the special choices that guarantee eventual termination, and their complete proof, are deferred to a longer treatment. The contribution recorded here is the automatic integer-valid cut construction and the algorithm announcement, rather than a complete proof of every cutting strategy or a modern mixed-integer solver.",
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    "title": "Gomory: cutting away a fractional answer without losing integer solutions",
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    "what_changed": "A mechanical fractional-tableau cut that preserves integer feasibility while excluding the fractional incumbent; a finite integer algorithm is announced, with its complete termination argument deferred.",
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