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.
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 J. Balogh J. Békési Gy. Dósa L. Epstein H. Kellerer Zs. Tuza Online results for black and white bin packing Theory Comput. Syst.56 1 (2015) 137–155. ⇒34
 N. Bansal Z. Liu A. Sankar Bin-packing with fragile objects and frequency allocation in cellular networks Wireless Networks15 6 (2009) 821–830. ⇒33
 M. Böhm J. Sgall P. Vesely Online colored bin packing arXiv:1404.5548 [cs.DS] (2014) ⇒34
 F. Clautiaux M. Dell’Amico M. Iori A. Khanafer Lower and upper bounds for the Bin Packing Problem with Fragile Objects Discrete Appl. Math.163 1 (2014) 73–86. ⇒33
5] Jr. E. G. Coffman C. Courcoubetis M. R. Garey D. S. Johnson P. W. Shor R. R. Weber M. Yannakakis Bin packing with discrete item sizes part I: Perfect packing theorems and the average case behavior of optimal packings SIAM J. Discrete Math.13 3 (2000) 384–402 ⇒45
 Jr. E. G. Coffman D. S. Johnson L. A. McGeoch R. R. Weber Bin packing with discrete item sizes part II: tight bounds on First Fit Random Structures Algorithms10 1–2 (1997) 69–101. ⇒45
 Gy. Dósa L. Epstein Colorful bin packing Algorithm Theory – SWAT 2014Lecture Notes in Comput. Sci.8503 (2014) 170–181. ⇒34
 Gy. Dósa J. Sgall First Fit bin packing: a tight analysis 30th International Symposium on Theoretical Aspects of Computer Science: STACS Dagstuhl Germany 2013 pp. 538–549. ⇒47
 Gy. Dósa Zs. Tuza D. Ye Bin packing with ’Largest In Bottom’ constraint: tighter bounds and generalizations J. Comb. Optim.26 3 (2013) 416–436. ⇒34
 L. Epstein On online bin packing with LIB Constraints Naval Res. Logist.56 8 (2009) 780–786. ⇒34
 L. Epstein Cs. Imreh A. Levin Class constrained bin packing revisited Theoret. Comput. Sci.411 34–36 (2010) 3073–3089. ⇒33
 L. Epstein A. Levin On bin packing with conflicts Approximation and Online AlgorithmsLecture Notes in Comput. Sci.4368 (2007) 160–731. ⇒34
 K. Jansen An approximation scheme for bin packing with conflicts J. Comb. Optim.3 4 (1999) 363–377. ⇒34
 K. Jansen S. Öhring Approximation algorithms for time constrained scheduling Inform. and Comput.132 2 (1997) 85–108. ⇒34
 A. Khanafer F. Clautiaux E. G. Talbi Tree-decomposition based heuristics for the two-dimensional bin packing problem with conflicts Computers and Operations Research39 1 (2012) 54–63. ⇒34
 P. Manyem Uniform sized bin packing and covering: Online version Topics in industrial mathematics Springer US 2000. ⇒34
 P. Manyem R. L. Salt M. S. Visser Approximation lower bounds in online LIB bin packing and covering J. Autom. Lang. Comb.8 4 (2003) 663–674 ⇒34
 S. Martello P. Toth Knapsack Problems: Algorithms and Computer Implementations John Wiley and Sons 1990. ⇒36
 B. McCloskey A. Shankar Approaches to bin packing with clique-graph conflicts EECS Department University of California Berkeley (2005) ⇒34
 R. Sadykov F. Vanderbeck Bin packing with conflicts: a generic branch-and-price algorithm INFORMS J. Comput.25 2 (2013) 244–255. ⇒34
 H. Shachnai T. Tamir Tight bounds for online class-constrained packing Theoret. Comput. Sci.321 1 (2004) 103–123. ⇒33 45
 H. Shachnai T. Tamir Polynomial time approximation schemes for class-constrained packing problems Journal of Scheduling4 6 (2001) 313–338. ⇒33
 H. Shachnai T. Tamir On two class-constrained versions of the multiple knapsack problem Algorithmica29 3 (2001) 442–467. ⇒33
 K. Thulasiraman M. N. S. Swamy 5.7 Acyclic Directed GraphsGraphs: Theory and Algorithms John Wiley and Sons 1992. 118. ⇒43
 J. D. Ullman The performance of a memory allocation algorithm. Princeton University Department of Electrical Engineering Computer Science Laboratory (1971) ⇒47
 E. C. Xavier F. K. Miyazawa The class constrained bin packing problem with applications to video-on-demand Theoret. Comput. Sci.393 1–3 (2008) 240–259. ⇒33