How to show that this problem is NP-hard: Find two subsets of 2 given sets such that the difference between the subset sums is $\leq v$

Question

As input, given two finite sets of integers $X = \{x_1,...,x_m\}$, $Y = \{y_1,...,y_n\} \subseteq Z$, and a non-negative integer $v ≥ 0$. The goal is to decide if there are non-empty subsets $S ⊆ [m]$ and $T ⊆ [n]$ such that

$$ \left|\sum_{i\in S}x_i-\sum_{j\in T} y_j\right| \leq v$$

How to show this problem is NP-hard? I'm quite confused. What I got so far is to reduce from subset sum problems, since the form is set to less than v. So I need to have 2v+1 subset sum problems to verify

Answer 1

The special case where $v=0$ and $Y=\{\frac12\sum_{i=1}^m x_i\}$ is the partition problem, which is NP-complete.