large and medium jobs efficiently we need to differentiate between bags that we are going to prioritize and bags that are less important. The property we want to enforce for such an important bag is that they hold a large amount of jobs with a certain large size. Therefore we introduce a notation for so called size-restricted bags. Further we define a functions to represent a sorted list of size-restricted bags of one size.

Let $B_l$ be any bag and $s$ any item size. With $B_l^s:= \{j\in B_l | p_j = s\}$ we denote the set of all jobs in $B_l$ with size $s$ and call $B_l^s$ *size-restricted* bag with size $s$. Also we define for each large item size $s$ a function $o_s:\mathbb{N}_{\le b} \Rightarrow \mathbb{N}_{\le b}$ to be a bijective index function such that for every $l< b$ we have that $|B^s_{o_s(l)}| \ge  |B^s_{o_s(l+1)}|$.

Given a size-restricted bag $B_l^s$ we may also refer to $B_l$ as the respective full bag. In order to find a feasible distribution of all large jobs we need to ensure that a constant number $b' \le b$ bags of each large item size is packed with no violations of the bag-constraints. Even if 