An exponential lower bound for Zadeh’s pivot rule
The question whether the Simplex Algorithm admits an efficient pivot rule remains one of the most important open questions in discrete optimization. While many natural, deterministic pivot rules are known to yield exponential running times, the random-facet rule was shown to have a subexponential ru...
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| Veröffentlicht in: | Mathematical programming 2023-05, Vol.199 (1-2), p.865-936 |
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| container_title | Mathematical programming |
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| creator | Disser, Yann Friedmann, Oliver Hopp, Alexander V. |
| description | The question whether the Simplex Algorithm admits an efficient pivot rule remains one of the most important open questions in discrete optimization. While many natural, deterministic pivot rules are known to yield exponential running times, the random-facet rule was shown to have a subexponential running time. For a long time, Zadeh’s rule remained the most prominent candidate for the first deterministic pivot rule with subexponential running time. We present a lower bound construction that shows that Zadeh’s rule is in fact exponential in the worst case. Our construction is based on a close relation to the Strategy Improvement Algorithm for Parity Games and the Policy Iteration Algorithm for Markov Decision Processes, and we also obtain exponential lower bounds for Zadeh’s rule in these contexts. |
| doi_str_mv | 10.1007/s10107-022-01848-x |
| format | Article |
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| subjects | Algorithms Calculus of Variations and Optimal Control Optimization Combinatorics Full Length Paper Iterative algorithms Lower bounds Markov processes Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Optimization Questions Run time (computers) Theoretical |
| title | An exponential lower bound for Zadeh’s pivot rule |
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