I have a set $ S$ of an even number of positive elements $ 2m$ and $ m$ values $ t_1,t_2,\ldots,t_m$ where each $ t_i\leq1$ for all $ i$ .

The question is: can you select $ m$ pairs $ (a_i,b_i)$ from $ S$ such that $ |a_i-b_i|\geq t_i$ ?

I was trying to prove that this problem is NP-hard by a reduction from 3-Partition Problem. I failed because if I choose the numbers as in 3-partition I cannot guarantee that their absolute difference is at least $ t_i$ .

Do you have any hints?