will be separated in two bags. One containing all large jobs and another one containing only small jobs. ]{} \[abb:modtra\]

\[lem:mod1\] Assume we are given an instances for machine-scheduling with bag-constraints $I$ and let $I'$ be the modified instance of $I$. If there is a solution for $I$ with makespan $C$ then there is also a solution for $I'$ with makespan $(1+\epsilon)*C$.

The advantage of this modification is that we only have non-priority bags that contain only large jobs or only small jobs. Thanks to this it is sufficient to schedule large and small jobs independently for these bags as they cannot conflict with each other. We can also see that we can apply this transformation in polynomial time and since we increase our number of jobs by at most factor $2$, this will not affect the overall running time of our algorithm except for this constant factor.

Now we will show how we can construct a solution with bounded height for the original instance given a solution for the modified one. First we will deal with the task to find a valid placement of medium jobs from non-priority bags as these were completely discarded in our modified instance. We 