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        "description": "The polynomial-time result fixes the number of integer variables. Its dimension-dependent cost is not a polynomial bound uniform in arbitrary dimension. Rational input encoding and geometric subroutines matter. Mixed variants have their own assumptions. The theorem is not evidence that every integer program is easy in practice.",
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      "statement": "# Lenstra: integer programming becomes tractable when dimension is fixed\n\nLenstra showed that integer linear feasibility can be solved in polynomial input-length time when the number of integer variables is fixed, using lattice geometry and reduction to lower-dimensional problems.\n\nAn optimization problem may have a feasible answer in real numbers but no answer in whole numbers. Integer programming captures this extra restriction, which is computationally hard in general. Lenstra’s 1983 paper proved that fixing the number of integer variables changes the complexity: the remaining input can grow while the problem stays polynomially solvable. The method reshapes the feasible region and uses lattice geometry to either find an integer point or split the search into a bounded number of lower-dimensional problems. Empty, unbounded and lower-dimensional regions are handled by reductions rather than silently excluded. The result also covers mixed-integer feasibility with a fixed number of integer variables. It does not make arbitrary large-dimensional integer programs easy: the original polynomial degree grows exponentially with dimension, and the paper describes its algorithms as primarily theoretical. Lovász’s algorithmic improvements, earlier low-dimensional work and the continuous linear-programming and lattice-reduction tools retain separate credit.\n\n## Scientific scope\n\nThe polynomial-time result fixes the number of integer variables. Its dimension-dependent cost is not a polynomial bound uniform in arbitrary dimension. Rational input encoding and geometric subroutines matter. Mixed variants have their own assumptions. The theorem is not evidence that every integer program is easy in practice.\n\n## People and systems\n\nH. W. Lenstra Jr.\n\n\n\n## Original sources\n\n- [Lenstra, Integer Programming with a Fixed Number of Variables, MOR8(4)538-548(1983), author-hosted journal copy](https://pub.math.leidenuniv.nl/~lenstrahw/PUBLICATIONS/1983i/art.pdf)\n- [Publisher Crossref deposit for10.1287/moor.8.4.538](https://api.crossref.org/works/10.1287%2Fmoor.8.4.538)\n",
      "summary": "Lenstra showed that integer linear feasibility can be solved in polynomial input-length time when the number of integer variables is fixed, using lattice geometry and reduction to lower-dimensional problems.\n\nAn optimization problem may have a feasible answer in real numbers but no answer in whole numbers. Integer programming captures this extra restriction, which is computationally hard in general. Lenstra’s 1983 paper proved that fixing the number of integer variables changes the complexity: the remaining input can grow while the problem stays polynomially solvable. The method reshapes the feasible region and uses lattice geometry to either find an integer point or split the search into a bounded number of lower-dimensional problems. Empty, unbounded and lower-dimensional regions are handled by reductions rather than silently excluded. The result also covers mixed-integer feasibility with a fixed number of integer variables. It does not make arbitrary large-dimensional integer programs easy: the original polynomial degree grows exponentially with dimension, and the paper describes its algorithms as primarily theoretical. Lovász’s algorithmic improvements, earlier low-dimensional work and the continuous linear-programming and lattice-reduction tools retain separate credit.",
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