small jobs, then we do not modify $B_l$ and continue with the next bag). Open up a new bag $B'_l$ that contains all large Jobs of $B_l$. And for all large and medium jobs $j \in B_l$ replace $j$ in $B_l$ with $\overline j$ such that $p_{\overline j} = p_{max}$. We will call these additionally added small jobs *filler-jobs*. Alter every non-priority bag in this way and we will end up with a modified instance that we denote with $I'$.

Intuitively we shortened all medium and large jobs to small jobs in non-priority bags, while saving copies of the large jobs in separate bags, as depicted in figure \[abb:modtra\]. Note that we removed medium jobs from non-priority bags. We will show how to use a result of Das and Wiese [@bagptas] to add these back to our instance. Before doing so we will first show that by applying this modification we overall lose only an $\epsilon$ factor in the objective.

[Example of a transformation of non-priority bag $B_l$. Jobs are coloured, large jobs are red, medium jobs yellow and small jobs green. Additionally filler-jobs are coloured blue to differ from normal small jobs. As seen in the picture each bag 