  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.
