Lenstra's most widely cited scientific result is the first polynomial time algorithm to LLL lattice reduction algorithm with Hendrik Willem Lenstra and L�szl?Lov�sz.
22.
This is weaker than saying it is a polynomial time algorithm, since it may run for super-polynomial time, but with very low probability.
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For theoretical purposes requiring a deterministic polynomial time algorithm, it was superseded by the AKS primality test, which also does not rely on unproven assumptions.
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Polynomial time algorithms are known for determining whether the derived graph of a \ mathbb { Z } ^ d-voltage graph contains any directed cycles.
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A major unresolved challenge is to determine if the problem admits a polynomial time algorithm, that is, whether the problem lies in the complexity class P.
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Description : There is a polynomial time algorithm to find a maximum matching in a graph that is not bipartite and another step toward the idea of computational complexity.
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Finally, subsequent to the proof of the strong perfect graph theorem, a polynomial time algorithm was discovered by Chudnovsky, Cornu�jols, Liu, Seymour, and Vuakovi.
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Then, Edmonds proved in the paper " Edge-Disjoint Branchings " the upper bound in the broadcast scenario is also achievable, and proposed a polynomial time algorithm.
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For any fixed m \ ge 2, these problems do admit a pseudo-polynomial time algorithm ( similar to the one for basic knapsack ) and a PTAS.
30.
By repeated random selection of a possible witness, the large probability that a random string is a witness gives an expected polynomial time algorithm for accepting or rejecting an input.
