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### Colored Bin Packing: Online Algorithms and Lower Bounds

Martin Böhm, György Dósa, Leah Epstein, Jiří Sgall, Pavel Veselý
2016 Algorithmica
For items of arbitrary size we give a lower bound of 2.5 on the asymptotic competitive ratio of any online algorithm and an absolutely 3.5-competitive algorithm.  ...  In the Colored Bin Packing problem a sequence of items of sizes up to 1 arrives to be packed into bins of unit capacity.  ...  Lower Bound for Items of Arbitrary Size We show a lower bound of 2 for two colors, i.e., for Black and White Bin Packing, and a lower bound of 2.5 for at least three colors.  ...

### Colorful Bin Packing [chapter]

György Dósa, Leah Epstein
2014 Lecture Notes in Computer Science
Our main results are a new algorithm for colorful bin packing that we design and analyze, whose absolute competitive ratio is 4, and a new lower bound of 2 on the asymptotic competitive ratio of any algorithm  ...  This problem generalizes standard online bin packing and online black and white bin packing (where |C| = 2).  ...  Lower bounds for arbitrary online algorithms are given in Section 3.  ...

### Colorful bin packing [article]

Gyorgy Dosa, Leah Epstein
2014 arXiv   pre-print
Our main results are a new algorithm for colorful bin packing that we design and analyze, whose absolute competitive ratio is 4, and a new lower bound of 2 on the asymptotic competitive ratio of any algorithm  ...  This problem generalizes standard online bin packing and online black and white bin packing (where |C|=2).  ...  Lower bounds for arbitrary online algorithms are given in Section 3.  ...

### Online Colored Bin Packing [article]

Martin Böhm, Jiří Sgall, Pavel Veselý
2014 arXiv   pre-print
In fact, the algorithm always uses at most 1.5· OPT bins and we show a matching lower bound of 1.5· OPT for any value of OPT≥ 2.  ...  In the case of two colors---the Black and White Bin Packing problem---we prove that all Any Fit algorithms have absolute competitive ratio 3.  ...  They improved the lower bound for online Black and White Bin Packing to 2 for deterministic algorithms, which holds for more colors as well.  ...

### Online Colored Bin Packing [chapter]

Martin Böhm, Jiří Sgall, Pavel Veselý
2015 Lecture Notes in Computer Science
In fact, the algorithm always uses at most 1.5 · OPT bins and we show a matching lower bound of 1.5 · OPT for any value of OPT ≥ 2.  ...  In particular, the absolute ratio of our algorithm is 5/3 and this is optimal. For items of arbitrary size we give a lower bound of 2.5 and an absolutely 3.5-competitive algorithm.  ...  They improved the lower bound for online Black and White Bin Packing to 2 for deterministic algorithms, which holds for more colors as well.  ...

### On Colorful Bin Packing Games [article]

Vittorio Bilò, Francesco Cellinese, Giovanna Melideo, Gianpiero Monaco
2017 arXiv   pre-print
Although, under both cost functions, colorful bin packing games do not converge in general to a (pure) Nash equilibrium, we show that Nash equilibria are guaranteed to exist and we design an algorithm  ...  Each item has one of m≥ 2 colors and cannot be packed next to an item of the same color.  ...  Competitive algorithms for the online colorful bin packing problem were presented in [11] .  ...

### A Two-Pass Algorithm for Unordered Colored Bin Packing

Ananya Christman, Hamza Alsarhan, Davin Chia, Shannia Fu, Yanfeng Jin
2016 International Conference on Discrete Optimization and Operations Research
In the Colored Bin Packing problem a set of items with varying weights and colors must be packed into bins of uniform weight limit such that no two items of the same color may be packed adjacently within  ...  We present exact, linear-time algorithms for this problem for the cases where there are two or more colors when the items have zero weight and when the items have unit weight.  ...  They prove upper and lower bounds for several variants of this problem in both the offline and online settings.  ...

### Selfish bin coloring

Leah Epstein, Sven O. Krumke, Asaf Levin, Heike Sperber
2010 Journal of combinatorial optimization
We introduce a new game, the so-called bin coloring game, in which selfish players control colored items and each player aims at packing its item into a bin with as few different colors as possible.  ...  We establish the existence of Nash and strong as well as weakly and strictly Pareto optimal equilibria in these games in the cases of capacitated and uncapacitated bins.  ...  For years the exact approximation ratio of this algorithm was unknown although almost matching upper and lower bounds were given by Caprara and Pferschy [5] .  ...

### VNS matheuristic for a bin packing problem with a color constraint

Y. Kochetov, A. Kondakov
2017 Electronic Notes in Discrete Mathematics
Each bin has a color capacity, the total number of colors for a bin is the union of colors for its items and can not exceed the bin capacity.  ...  We study a new variant of the bin packing problem. Given a set of items, each item has a set of colors.  ...  Columns LB and UB present the lower bound of the linear programming relaxation and the best upper bound for the global optimum.  ...

### Strip Packing vs. Bin Packing [article]

Xin Han, Kazuo Iwama, Deshi Ye, Guochuan Zhang
2006 arXiv   pre-print
In this paper we establish a general algorithmic framework between bin packing and strip packing, with which we achieve the same asymptotic bounds by applying bin packing algorithms to strip packing.  ...  It implies online strip packing admits an upper bound of 1.58889 on the asymptotic competitive ratio, which is very close to the lower bound 1.5401 and significantly improves the previously best bound  ...  More precisely, strip packing is NP-hard in the strong sense and the lower bound 1.5401 [15] is valid for online strip packing. Previous results. For the offline version Coffman et al.  ...

### Improved Online Hypercube Packing [chapter]

Xin Han, Deshi Ye, Yong Zhou
2007 Lecture Notes in Computer Science
Based on the techniques in one dimensional bin packing algorithm Super Harmonic by Seiden, we give a new framework for online hypercube packing and obtain new upper bounds of asymptotic competitive ratios  ...  In this paper, we study online multidimensional bin packing problem when all items are hypercubes.  ...  On online hypercube packing, Coppersmith and Raghavan [3] showed an upper bound of 43/16 = 2.6875 for online square packing and an upper bound 6.25 for online cube packing.  ...

### Improved online hypercube packing [article]

Xin Han, Deshi Ye, Yong Zhou
2006 arXiv   pre-print
Based on the techniques in one dimensional bin packing algorithm Super Harmonic by Seiden, we give a framework for online hypercube packing problem and obtain new upper bounds of asymptotic competitive  ...  In this paper, we study online multidimensional bin packing problem when all items are hypercubes.  ...  On online hypercube packing, Coppersmith and Raghavan [3] showed an upper bound of 43/16 = 2.6875 for online square packing and an upper bound 6.25 for online cube packing.  ...

### Black and White Bin Packing Revisited [chapter]

Jing Chen, Xin Han, Wolfgang Bein, Hing-Fung Ting
2015 Lecture Notes in Computer Science
lower bound 1.5 Xin Han, Black And White Bin Packing Revisited upper bound 1.5 Online Algorithm, and c ≥ 3 upper bound Xin Han, Black And White Bin Packing Revisited  ...  Xin Han, Black And White Bin Packing Revisited B-W Bin packing Difficulties New techniques Online bin packing Previous results Lower bounds: 1.5 → 1.54017 → 1.54037, [2012TCS].  ...

### Class constrained bin packing revisited

Leah Epstein, Csanád Imreh, Asaf Levin
2010 Theoretical Computer Science
We study the following variant of the bin packing problem. We are given a set of items, where each item has a (non-negative) size and a color.  ...  1 (as in the classical bin packing problem) and the total number of colors of the items in S is at most k (which distinguishes our problem from the classical version).  ...  Lower bounds for online algorithms and k = 2 In this section we provide lower bounds on the performance guarantees of specific algorithms as well as a lower bound on the performance of any online algorithm  ...

### Dynamic multi-dimensional bin packing

Leah Epstein, Meital Levy
2010 Journal of Discrete Algorithms
This algorithm was studied before for rectangle packing and for square packing and was generalized only for multi-dimensional cubes. We present upper and lower bounds for each of these cases.  ...  A natural generalization of the classical online bin packing problem is the dynamic bin packing problem introduced by Coffman et al. (1983) [7].  ...  In each part, items of the same size and type are given and depart. Using a computer we get for n = 10 000, a lower bound of 4.85383. 2  ...
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