Finding suitable NP-complete problem to reduce to my problem
This problem is essentially just a rephrasing of the classic NP-complete problem PARTITION. If you can partition a set $S$ of natural numbers into two partitions $S_1$ and $S_2$ with equal sum, then $\sum_{s \in S_1} s + \sum_{s \in S_2} -s = 0$ and vice versa.