Input
We are given a set of basis elements, $\ v_1$,$\ v_2$ ,...,$\ v_n$ of a $\mathbb Z^m$- module and a multiset of integers $\ B :=$ {$\ b_1, ..., b_m$}
Desired Output
Return true if there exists a linear combination $\ a_1v_1 + a_2v_2 + ... + a_nv_n$ such that the resulting vector uses the integers $\ b_1, ... , b_m $ the same number of times that each occurs in B. Return false otherwise.
Example
Let $\ v_1 = (1,0,0,1), v_2 = (1,0,1,0) $ and $\ B =$ {$\ 0,4,2,2$}. Here the algorithm should return true since we can write $\ (4,0,2,2)$ as $\ 2v_1 + 2v_2$.
As another example, consider the same vectors with the integers {1,2,3,4}. Here we want to return false since we will never be able to write four strictly positive numbers as a linear combination of the two vectors.
So my questions are: What kind of problem is this? Can anyone prove it NP-complete or find a reasonable algorithm for it?