On Probabilistic Algorithm for Solving Almost All Instances of the Set Partition Problem
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Earlier, I.V. Latkin and the author have shown the set partition problem can be reduced to the problem of finding singular points of a cubic hypersurface. The article focuses on the new link between two different research areas as well as on methods to look for singular points or to confirm the smoothness of the hypersurface. Our approach is based on the description of tangent lines to the hypersurface. The existence of at least one singular point imposes a restriction on the algebraic equation that determines the set of tangent lines passing through the selected point of the space. This equation is based on the formula for the discriminant of a univariate polynomial. We have proposed a probabilistic algorithm for some set of inputs of the set partition problem. The probabilistic algorithm is not proved to have polynomial complexity.
KeywordsSet partition Cubic hypersurfaces Smoothness Tangent line Polynomial Discriminant Computational complexity
The author would like to thank Mark Spivakovsky, Sergei P. Tarasov, Mikhail N. Vyalyi, and the anonymous reviewers for useful comments.
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