# 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 $\leq v$

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

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