Improved smoothed analysis of the shadow vertex simplex method

Spielman and Teng (JACM '04), proved that the smoothed complexity of a two-phase shadow-vertex method for linear programming is polynomial in the number of constraints n, the number of variables d, and the parameter of perturbation 1/sigma. The key geometric result in their proof was an upper bound of O(nd(3)/min (sigma,(9dln n)(-1/2))(6)) on the expected size of the shadow of the polytope defined by the perturbed linear program. In this paper, we give a much simpler proof of a better bound: When evaluated at sigma = (9dln n)(-1/2) this improves the size estimate from O(nd(6)In(3) n) to O(n(2) d(2) ln n). The improvement only becomes better as sigma decreases. The bound on the running time of the two-phase shadow vertex proved by Spielman and Teng is dominated by the exponent of sigma in the shadow-size bound. By reducing this exponent from 6 to 2, we decrease the exponent in the smoothed complexity of the two-phase shadow vertex method by a multiplicative factor of 3.