On polynomial-time solvable linear Diophantine problems

Abstract

We obtain a polynomial-time algorithm that, given input (A, b), where A=(B|N) is an integer mxn matrix, m<n, with nonsingular mxm submatrix B and b is an m-dimensional integer vector, 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.

This is a generalisation and extension of the results from the previous version. At the time the previous version was written, the author was not aware of the results of Brimkov [5]