Relation with the perfect partition problem and the single machine task schedule problem Perfect partition problem (PPP): given $x_1,...,x_n\in \mathbb{N}$ we want to know if there's a set $S\subset \{1,2,...,n\}$ such that $$\sum_{i\in S}x_i=\sum_{j\notin S}x_j.$$

Single machine task schedule problem (SMP): a machine can perform only one task $T_i$ with duration $d_i$ at a time and we want to know if all tasks can be done, given that $T_i$ can't start before the time $a_i$ or be finished after the time $b_i$ with $i\in \{1,2,3,...,n\}$.

I want to know how to reduce PPP to SMP.

EDIT: I tried modelling SMP using the constraints and the existence of a bijection $f:\{1,2,3,...,n\} \rightarrow \{1,2,3,...,n\}$ such that $$a_{f(i)}\leq \sum_{j=1}^i d_{f(j)}\leq b_{f(i)}$$ for all $i\in \{1,2,3,...,n\}$. And I tried to make $x_i=d_i$ and get the conditions of the SMP to make the sum be equal to que complementary $\left (\frac{\sum_{i=1}^{n} x_i}{2}\right)$ at some point, but I got stuck as I can't find how to relate $f$ to $S$ from the PPP, that's not really working.
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.