An Application of Integer Programming to the Decomposition of Numerical Semigroups

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.