Relation with the perfect partition problem and the single machine task schedule problem
Just let $d_i=x_i$, $a_1=\frac{\sum_{i=1}^n x_i}{2}-d_1$ and $b_1=\frac{\sum_{i=1}^n x_i}{2}$, and $a_i=0$ and $b_i=\sum_{i=1}x_i$ for other $i$'s. Then easy to see there is a solution for the original PPP instance iff there is a solution for this SMP instance.