Beyond Hirsch Conjecture: Walks on Random Polytopes and Smoothed Complexity of the Simplex Method

• Authors: Roman Vershynin
• Venue: FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
• Year: 2006

Spielman and Teng proved that the shadow-vertex simplex method had polynomial smoothed complexity. On a slight random perturbation of arbitrary linear program, the simplex method finds the solution after a walk on the feasible polytope(s) with expected length polynomial in the number of constraints n, the number of variables d and the inverse standard deviation of the perturbation 1/s. We show that the length of walk is actually polylogarithmic in the number of constraints n. We thus improve Spielman-Teng's bound on the walk O*(n^{86} d^{55} \sigma ^{ - 30} ) to O(max(d^{5} log^2 n, d^9 log^4 d, d^{3}\sigma^{-4})). This in particular shows that the tight Hirsch conjecture n - d on the diameter of polytopes is not a limitation for the smoothed Linear Programming. Random perturbations create short paths between vertices. We propose a randomized phase-I for solving arbitrary linear programs. Instead of finding a vertex of a feasible set, we add a vertex at random to the feasible set. This does not affect the solution of the linear program with constant probability. So, in expectation it takes a constant number of independent trials until a correct solution is found. This overcomes one of the major difficulties of smoothed analysis of the simplex method -- one can now statistically decouple the walk from the smoothed linear program. This yields a much better reduction of the smoothed complexity to a geometric quantity -- the size of planar sections of random polytopes. We also improve upon the known estimates for that size.