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October 17, 2025TheoretiCSOpen Access

Upper and Lower Bounds on the Smoothed Complexity of the Simplex Method

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New analysis reveals improved upper bounds on smoothed complexity in linear programming, indicating advancements in theory.

The smoothed complexity of the simplex method is improved to O(σ^{-3/2} d^{13/4} log^{7/4} n), indicating better efficiency.
A lower bound of Ω(min(σ^{-1/2} d^{-1/2} log^{-1/4} d, 2^d)) pivot steps highlights the complexity of the shadow vertex simplex method.
The analysis utilizes a novel variation on the extended formulation for regular 2^k-gons, showcasing its importance in complexities.
Numerical experiments suggest further improvements may be achievable in the smoothed complexity analysis.

Huiberts et al. (2025) studied this question.

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