On polynomial-time solvable linear Diophantine problems

Abstract

We obtain a polynomial-time algorithm that, given input $\left(A,b\right)$, where $A=\left(B|N\right)\in {\mathbb{Z}}^{m\times n}$, $m<n$, with nonsingular $B\in {\mathbb{Z}}^{m\times m}$ and $b\in {\mathbb{Z}}^m$, finds a nonnegative integer solution to the system $Ax=b$ or determines that no such solution exists, provided that $b$ is located sufficiently “deep” in the cone generated by the columns of $B$. This result improves on some of the previously known conditions that guarantee polynomial-time solvability of linear Diophantine problems.