Bin packing with directed stackability conflicts

The Bin Packing problem is a well-known and highly investigated problem in the computer science: we have n items given with their sizes, and we want to assign them to unit capacity bins such, that we use the minimum number of bins.

In this paper, some generalizations of this problem are considered, where there are some additional stackability constraints defining that certain items can or cannot be packed on each other. The corresponding model in the literature is the Bin Packing Problem with Conflicts (BPPC), where this additional constraint is defined by an undirected conflict graph having edges between items that cannot be assigned to the same bin. However, we show some practical cases, where this conflict is directed, meaning that the items can be assigned to the same bin, but only in a certain order. Two new models are introduced for this problem: Bin Packing Problem with Hanoi Conflicts (BPPHC) and Bin Packing Problem with Directed Conflicts (BPPDC). In this work, the connection of the three conflict models is examined in detail.

We have investigated the complexity of the new models, mainly the BPPHC model, in the special case where each item have the same size. We also considered two cases depending on whether re-ordering the items is allowed or not.

We show that for the online version of the BPPHC model with unit size items, every Any-Fit algorithm gives not better than -competitive, when it is forbidden for the optimum to re-order the items, even if only 2 stackability classes, called Hanoi classes, are applied. This lower bound is generalized for arbitrary number of Hanoi classes. However, we also prove, that asymptotically the First-Fit algorithm is 1-competitive for this case.

Finally, we introduce an algorithm for the offline version of the BPPHC model with unit size items, which has polynomial time complexity, if the number of the Hanoi classes and the capacity of the bins are constant.