Las Vegas algorithms for linear and integer programming when the dimension is small

This paper gives an algorithm for solving linear programming problems. For a problem with n constraints and d variables, the algorithm requires an expected<inline-equation><f>O<fen lp="par">d<sup>2</sup>n<rp post="par"></fen>+<fen lp="par">logn<rp post="par"></fen>O<fen lp="par">d<rp post="par"></fen><sup>d/2+O<fen lp="par">1<rp post="par"></fen></sup>+O<fen lp="par">d<sup>4</sup><rad><rcd>n</rcd></rad>logn<rp post="par"></fen></f></inline-equation> arithmetic operations, as<inline-equation><f>n→∞</f></inline-equation>. The constant factors do not depend on d. Also, an algorithm is given for integer linear programming. Let <inline-equation><f><g>4</g></f></inline-equation> bound the number of bits required to specify the rational numbers defining an input constraint or the objective function vector. Let n and d be as before. Then, the algorithm requires expected <inline-equation> <f> O<fen lp="par">2<sup>d</sup>dn+8<sup>d</sup>d<rad><rcd>n<rm>l n<it>n<rm></rm></it></rm></rcd></rad><rm>ln<it>n</it></rm><rp post="par"></fen> +d<sup>o<fen lp="par">d<rp post="par"></fen></sup><g>4</g><rm> ln<it>n</it></rm></f> </inline-equation> operations on numbers with <inline-equation> <f> d<sup>o<fen lp="par">1<rp post="par"></fen></sup><g>4</g></f> </inline-equation> bits, as <inline-equation> <f> n→∞</f> </inline-equation>, where the constant factors do not depend on d or<inline-equation><f><g>4</g></f></inline-equation>to other convex programming problems. For example, an algorithm for finding the smallest sphere enclosing a set of n points in <inline-equation><f>E<sup>d</sup></f></inline-equation>has the same time bound.<?Pub Caret>

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