Computation of atomic fibers of Z-linear maps

For given matrix $Ain^{d imes n}$, the set $P_{b}={z:Az=b,zin^n_+}$ describes the preimage or fiber of $bin^d$ under the $$-linear map $f_A:^n_+ ightarrow^d$, $xmapsto Ax$. The fiber $P_{b}$ is called atomic, if $P_{b}=P_{b_1}+P_{b_2}$ implies $b=b_1$ or $b=b_2$. In this paper we present a novel algorithm to compute such atomic fibers. An algorithmic solution to appearing subproblems, computational examples and applications are included as well.