Mathematical Programming: Series A and B
Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
Journal of the ACM (JACM)
A randomized polynomial-time simplex algorithm for linear programming
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Smoothed analysis of binary search trees
Theoretical Computer Science
Smoothed analysis of probabilistic roadmaps
Computational Geometry: Theory and Applications
Algorithm engineering: bridging the gap between algorithm theory and practice
Algorithm engineering: bridging the gap between algorithm theory and practice
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Spielman and Teng (JACM 驴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 ,(9d\ln n)^{ - 1/2} )^6 ) on the expected size of the shadow of the polytope de?ned by the perturbed linear program. In this paper, we give a much simpler proof of a better bound: o(n^3 d\ln n/\min (\sigma ,(4d\ln n)^{ - 1/2} )^2 )When evaluated at \sigma= (9d\ln n)^{ - 1/2}, this improves the size estimate from O(nd^6 \ln ^3 n) to o(n^2 d^2 \ln n). The improvement only becomes better as s decreases.