machine, as these jobs were separated in two bags in $I'$. Using filler-jobs from $I'$ we can fix these conflicts and therefore generate a solution for $I$.

\[lem:mod3\] Let $I$ be an instance of the machine scheduling with bag-constraint problem and let $I'$ be the modified instance. Let $S'$ be a solution of the instance $I'$ that additionally contains all medium jobs from non-priority bags of instance $I$ as per lemma \[lem:mod2\]. Then we can find a valid solution $S$ for instance $I$ that has at most the same makespan as $I'$.

With the last two lemmas we conclude that our modification not only gives us an easier structure to work with, but also keeps the overall error small. For the following we therefore assume that we continue working with our modified instance $I'$ and will denote this with $I$ in the next sections. Due to the modification and lemma \[lem:mod1\] we know that our estimated optimal value for this instance increases up to $T = 1+2\epsilon+\epsilon^2$ (remember that our previous optimum was $1+\epsilon$ due to rounding).

For the rest of this paper we will show how to compute a $1+O(\epsilon)$ for our modified instance. With the last two lemmas 