Gr{\"o}bner basis. a "pseudo-polynomial" algorithm for computing the Frobenius number

TL;DR

This paper introduces a pseudo-polynomial algorithm for computing the Frobenius number and Apéry set of a numerical semigroup generated by natural numbers, with applications to the integer knapsack problem, using Gröbner basis techniques without Buchberger's algorithm.

Contribution

It presents a novel pseudo-polynomial algorithm for Frobenius number computation and related structures, avoiding traditional Gröbner basis methods.

Findings

Algorithm computes Frobenius number and Apéry set efficiently.

Solves the integer knapsack problem in pseudo-polynomial time.

Provides a Gröbner basis construction without Buchberger's algorithm.

Abstract

Let consider $n$ natural numbers $a\_1 ,\ldots , a\_{n} $. Let $S$ be the numerical semigroup generated by $a\_1 ,\ldots , a\_{n} $. Set $A=K[t^{a\_1}, \ldots , t^{a\_n}]=K[{x\_1}, \ldots , {x\_n}]/I$. The aim of this paper is: \begin{enumerate}\item Give an effective pseudo-polynomial algorithm on $a\_1$, which computes The Ap{\'e}ry set and the Frobenius number of $S$. As a consequence it also solves in pseudo-polynomial time the integer knapsack problem : given a natural integer b, b belongs to $S$?\item The \gbb of $I$ for the reverse lexicographic order to $x\_n,\ldots ,x\_1$, without using Buchberger's algorithm. \item $\ini{I} $ for the reverse lexicographic order to $x\_n,\ldots ,x\_1$.\item $A$ as a $K[t^{ a\_1 }]$-module. \end{enumerate} We dont know the complexity of our algorithm. We need to solve the "multiplicative" integer knapsack problem: Find all positive integer solutions $({k\_1}, \ldots , {k\_n})$ of the inequality $\prod\_{i=2}^n (k\_i+1)\leq a\_1+1$. This algorithm is easily implemented. The implementation of this algorithm "frobenius-number-mm", for $n=17 $, can be downloaded in \hfill\breakhttps://www-fourier.ujf-grenoble.fr/~morales/frobenius-number-mm