A combinatorial bound for linear programming and related problems

We present a simple randomized algorithm which solves linear programs with n constraints and d variables in expected O(d32dn) 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.