Optimal Smoothed Analysis of the Simplex Method
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866913158438846464 |
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| author | Bach, Eleon Huiberts, Sophie |
| author_facet | Bach, Eleon Huiberts, Sophie |
| contents | Smoothed analysis is a method for analyzing the performance of algorithms, used especially for those algorithms whose running time in practice is significantly better than what can be proven through worst-case analysis. Spielman and Teng (STOC '01) introduced the smoothed analysis framework of algorithm analysis and applied it to the simplex method. Given an arbitrary linear program with $d$ variables and $n$ inequality constraints, Spielman and Teng proved that the simplex method runs in time $O(σ^{-30} d^{55} n^{86})$, where $σ> 0$ is the standard deviation of Gaussian distributed noise added to the original LP data. Spielman and Teng's result was simplified and strengthened over a series of works, with the current strongest upper bound being $O(σ^{-3/2} d^{13/4} \log(n)^{7/4})$ pivot steps due to Huiberts, Lee and Zhang (STOC '23). We prove that there exists a simplex method whose smoothed complexity is upper bounded by $O(σ^{-1/2} d^{11/4} \log(n)^{7/4})$ pivot steps. Furthermore, we prove a matching high-probability lower bound of $Ω( σ^{-1/2} d^{1/2}\ln(4/σ)^{-1/4})$ on the combinatorial diameter of the feasible polyhedron after smoothing, on instances using $n = \lfloor (4/σ)^d \rfloor$ inequality constraints. This lower bound indicates that our algorithm has optimal noise dependence among all simplex methods, up to polylogarithmic factors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_04197 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Optimal Smoothed Analysis of the Simplex Method Bach, Eleon Huiberts, Sophie Data Structures and Algorithms Smoothed analysis is a method for analyzing the performance of algorithms, used especially for those algorithms whose running time in practice is significantly better than what can be proven through worst-case analysis. Spielman and Teng (STOC '01) introduced the smoothed analysis framework of algorithm analysis and applied it to the simplex method. Given an arbitrary linear program with $d$ variables and $n$ inequality constraints, Spielman and Teng proved that the simplex method runs in time $O(σ^{-30} d^{55} n^{86})$, where $σ> 0$ is the standard deviation of Gaussian distributed noise added to the original LP data. Spielman and Teng's result was simplified and strengthened over a series of works, with the current strongest upper bound being $O(σ^{-3/2} d^{13/4} \log(n)^{7/4})$ pivot steps due to Huiberts, Lee and Zhang (STOC '23). We prove that there exists a simplex method whose smoothed complexity is upper bounded by $O(σ^{-1/2} d^{11/4} \log(n)^{7/4})$ pivot steps. Furthermore, we prove a matching high-probability lower bound of $Ω( σ^{-1/2} d^{1/2}\ln(4/σ)^{-1/4})$ on the combinatorial diameter of the feasible polyhedron after smoothing, on instances using $n = \lfloor (4/σ)^d \rfloor$ inequality constraints. This lower bound indicates that our algorithm has optimal noise dependence among all simplex methods, up to polylogarithmic factors. |
| title | Optimal Smoothed Analysis of the Simplex Method |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2504.04197 |