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//spl sigma/. The key geometric result in their proof was an upper bound of O(nd/sup 3//min (/spl sigma/, (9d ln n)/sup 1/2 /)/sup 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(n/sup 2/ d ln n/min (/spl sigma/, (4d ln n)/sup 1/2 /)/sup 2/). When evaluated at /spl sigma/ = (9d ln n)/sup 1/2 /, this improves the size estimate from O(nd/sup 6/ ln/sup 3/ n) to O(n/sup 2/d/sup 2/ ln n). The improvement only becomes better as /spl 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 /spl 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.