Lecture Notes in Computer Science Volume 577, 1992, pp 567-579

A combinatorial bound for linear programming and related problems

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We present a simple randomized algorithm which solves linear programs with n constraints and d variables in expected O(d 32 d n) time. The expectation is over the internal randomizations performed by the algorithm, and holds for any input.

The algorithm is presented in an abstract framework, which facilitates its application to a large class of problems, including computing smallest enclosing balls (or ellipsoids) of finite point sets in d-space, computing largest balls (ellipsoids) in convex polytopes, convex programming in general, etc.