On probabilistic algorithm for solving almost all instances of the set partition problem

Abstract

Earlier, I.V. Latkin and the author have shown the set par-tition 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 exis-tence 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 probabilis-tic algorithm for some set of inputs of the set partition problem. The probabilistic algorithm is not proved to have polynomial complexity.