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

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/sigma. 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*(n⁸⁶d ⁵⁵sigma^-30) to O(max(d⁵log²n, d⁹ log⁴d, d ³sigma^-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