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/σ. The key geometric result in their proof was an upper bound of O(nd³/min (σ, (9d ln n)¹2 /)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: O(n² d ln n/min (σ, (4d ln n)¹2 /)²). When evaluated at σ = (9d ln n)¹2 /, this improves the size estimate from O(nd⁶ ln³ n) to O(n²d² ln n). The improvement only becomes better as σ decreases. The bound on the running time of the two-phase shadow vertex proved by Spielman and Teng is dominated by the exponent of σ 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.