Random walks on polytopes and an affine interior point method for linear programming

Let K be a polytope in Rn defined by m linear inequalities. We give a new Markov Chain algorithm to draw a nearly uniform sample from K. The underlying Markov Chain is the first to have a mixing time that is strongly polynomial when started from a "central" point x0. If s is the supremum over all chords pq passing through x0 of |p-x0| |q-x0| and is an upper bound on the desired total variation distance from the uniform, it is sufficient to take O mn n log(sm) + log 1 steps of the random walk. We use this result to design an affine interior point algorithm that does a single random walk to solve linear programs approximately. More precisely, suppose Q = {z Bz 1} contains a point z such that cT z d and r := supzQ Bz + 1, where B is an m ? n matrix. Then, after = O mn n ln mr + ln 1 steps, the random walk is at a point x for which cT x d(1 - ) with probability greater than 1 - . The fact that this algorithm has a run-time that is provably polynomial is notable since the ana...

• Type: Conference
• Year: 2009
• Where: STOC
• Authors: Ravi Kannan, Hariharan Narayanan
• Keywords: Algorithms, Deterministic Affine Algorithm, Interior Point Algorithm, Markov Chain Algorithm, STOC 2009