just like in this definition. Intuitively we look at all size restricted bags (for large item sizes) and take the first $b'$ bags in the sorted lists respectively and call these and their respective full bags priority bags. Note that we can add large bags to the set of priority bags as their number is also bounded by $O(\frac{1}{\epsilon^{k+2}})$. This gives us the advantage that all non-priority bags are small bags. We will further modify small bags to ensure we can place small jobs and large jobs from non-priority bags independent from each other.

Instance Transformation
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The first step of our algorithm is to make further modifications on the instance. In order to make placing non-priority bags easier, we will split large and small jobs in separate bags. We further add small jobs for every medium and large job, so that after finding a solution for this modified instance, we also can revert the modifications to get a solution for the instance with the original bags.

Consider therefore the following process: Let $B_l$ be any non-priority bag and $p_{max}$ the height of the highest job that is still only a small job in $B_l$ (if $B_l$ should hold no 