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Minkowski’s convex body theorem and integer programming. (English) Zbl 0639.90069
The paper presents an algorithm for solving integer programming problems whose running time depends on the number n of variables as \(n^{O(n)}\). This is done by reducing an n variable problem to \((2n)^{5i/2}\) problems in n-i variables for some i greater than zero chosen by the algorithm. The factor of \(O(n^{5/2})\) “per variable” improves the best previously known factor which is exponential in n. Minkowski’s Convex Body theorem and other results from the geometry of numbers play a crucial role in the algorithm. Several related algorithms for lattice problems are presented. The complexity of these problems with respect to polynomial-time reducibilities is studied.

MSC:
90C10 Integer programming
52Bxx Polytopes and polyhedra
68Q25 Analysis of algorithms and problem complexity
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