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In 2018, it was proved that a particular variant of the simplex method is [[NP-mighty]], i.e., it can be used to solve, with polynomial overhead, any problem in NP implicitly during the algorithm's execution. Moreover, deciding whether a given variable ever enters the basis during the algorithm's execution on a given input, and determining the number of iterations needed for solving a given problem, are both [[NP-hardness|NP-hard]] problems.<ref>{{Cite journal|last=Disser|first=Yann|last2=Skutella|first2=Martin|date=2018-11-01|title=The Simplex Algorithm Is NP-Mighty|journal=ACM Trans. Algorithms|volume=15|issue=1|pages=5:1–5:19|doi=10.1145/3280847|issn=1549-6325|arxiv=1311.5935}}</ref> Computing the output of some other pivot rules was already known to be [[PSPACE-complete]]<ref>{{Citation | last = Adler | first = Ilan | last2 = Christos | first2 = Papadimitriou | author2-link = Christos Papadimitriou | last3 = Rubinstein | first3 = Aviad | title = On Simplex Pivoting Rules and Complexity Theory | journal = International Conference on Integer Programming and Combinatorial Optimization | volume = 17 | pages = 13-24 | year = 2014 | arxiv = 1404.3320 | doi = 10.1007/978-3-319-07557-0_2}}</ref><ref>{{Citation | last = Fearnly | first = John | last2 = Savani | first2 = Rahul | title = The Complexity of the Simplex Method | journal = Proceedings of the forty-seventh annual ACM symposium on Theory of computing | pages = 201-208 | year = 2015 | arxiv = 1404.0605 | doi = 10.1145/2746539.2746558}}</ref>.
Analyzing and quantifying the observation that the simplex algorithm is efficient in practice despite its exponential worst-case complexity has led to the development of other measures of complexity. The simplex algorithm has polynomial-time [[Best, worst and average case|average-case complexity]] under various [[probability distribution]]s, with the precise average-case performance of the simplex algorithm depending on the choice of a probability distribution for the [[random matrix|random matrices]].<ref name="Schrijver" /><ref name="Borgwardt">The simplex algorithm takes on average ''D'' steps for a cube. {{harvtxt|Borgwardt|1987}}: {{cite book|last=Borgwardt|first=Karl-Heinz|title=The simplex method: A probabilistic analysis|series=Algorithms and Combinatorics (Study and Research Texts)|volume=1|publisher=Springer-Verlag|___location=Berlin|year=1987|pages=xii+268|isbn=978-3-540-17096-9|mr=868467|ref=harv}}</ref> Another approach to studying "[[porous set|typical phenomena]]" uses [[Baire category theory]] from [[general topology]], and to show that (topologically) "most" matrices can be solved by the simplex algorithm in a polynomial number of steps.{{Citation needed|date=June 2019}} Another method to analyze the performance of the simplex algorithm studies the behavior of worst-case scenarios under small perturbation – are worst-case scenarios stable under a small change (in the sense of [[structural stability]]), or do they become tractable? Formally, this method uses fixed problems to which is added an amount of random noise, generalizing certain average-case models ("[[smoothed complexity]]").<ref>{{Cite book | last1=Spielman | first1=Daniel | last2=Teng | first2=Shang-Hua | author2-link=Shanghua Teng | title=Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing | publisher=ACM | isbn=978-1-58113-349-3 | doi=10.1145/380752.380813 | year=2001 | chapter=Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time| pages=296–305 | arxiv=cs/0111050}}</ref>
==Other algorithms==
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