Integer Programming and m-irreducibility of numerical semigroups

TL;DR

This paper develops polynomial-time algorithms for decomposing numerical semigroups into m-irreducible components using integer linear programming and introduces Kunz-coordinates to facilitate the process.

Contribution

It translates the algebraic problem into discrete optimization problems and provides a polynomial-time algorithm for finding minimal m-irreducible decompositions.

Findings

Polynomial-time algorithms for m-irreducible decomposition.

Effective use of Kunz-coordinates in optimization.

Computational experiments demonstrate efficiency.

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.