Gröbner basis. a "pseudo-polynomial" algorithm for computing the Frobenius number

Marcel Morales (1) , Dung Nguyen Thi (2)

1 IF - Institut Fourier
2 Thai Nguyen University of Agriculture and Foresty

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 Apéry 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

Mots clés: Algorithm; Apery set; Frobenius number; Grobner basis

Domaines: Algèbre commutative [math.AC]; Combinatoire [math.CO]

Marcel Morales, Dung Nguyen Thi. Gröbner basis. a "pseudo-polynomial" algorithm for computing the Frobenius number. 2015. ⟨hal-01212986v2⟩