A Subexponential Bound for Linear Programming

We present a simple randomized algorithm which solves linear programs with n constraints and d variables in expected min{O(d²2^dn), e^{2√d ln(n√d)+O(√d+ln n}} time in the unit cost model (where we count the number of arithmetic operations on the numbers in the input); to be precise, the algorithm computes the lexicographically smallest nonnegative point satisfying n given linear inequalities in d variables. The expectation is over the internal randomizations performed by the algorithm, and holds for any input. In conjunction with Clarkson's linear programming algorithm, this gives an expected bound of O(d²n + e^{O(√d ln d)}). The algorithm is presented in an abstract framework, which facilitates its application to several other related problems like computing the smallest enclosing ball (smallest volume enclosing ellipsoid) of n points in d-space, computing the distance of two n-vertex (or n-facet) polytopes in d-space, and others. The subexponential running time can also be established for some of these problems (this relies on some recent results due to Gärtner).