Assuming $x(e)=1$ in Condition 2, the problem is NP-complete.
Clearly it is in NP. We show NP-hardness by reduction from Subset Sum:
Lemma 1. The problem is NP-hard.
Proof. The proof is by the following reduction from Subset Sum. Given a Subset-Sum input $(y, T)$, where $y=(y_1, y_2, \ldots, y_n)$ is a sequence of integers, and $T$ is the target, the ...