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arXiv · 1101.4112

Integer Programming and m-irreducibility of numerical semigroups

Víctor Blanco·Justo Puerto
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Abstract

This paper addresses the problem of decomposing a numerical semigroup into m-irreducible numerical semigroups. The problem originally stated in algebraic terms is translated, introducing the so called Kunz-coordinates, to resolve a series of several discrete optimization problems. First, we prove that finding a minimal m-irreducible decomposition is equivalent to solve a multiobjective linear integer problem. Then, we restate that problem as the problem of finding all the optimal solutions of a finite number of single objective integer linear problems plus a set covering problem. Finally, we prove that there is a suitable transformation that reduces the original problem to find an optimal solution of a compact integer linear problem. This result ensures a polynomial time algorithm for each given multiplicity m. We have implemented the different algorithms and have performed some computational experiments to show the efficiency of our methodology.

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Víctor Blanco, Justo Puerto. 2011-01-21. Integer Programming and m-irreducibility of numerical semigroups. https://arxiv.org/abs/1101.4112

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