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 $m$ disjoint 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?