Colorful Bin Packing

Dósa, György; Epstein, Leah

doi:10.1007/978-3-319-08404-6_15

Cited by 9 publications

(7 citation statements)

References 12 publications

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“…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 (see Section 3.1). This is significantly stronger than the asymptotic lower bound of 1.5 of Dósa and Epstein [8], in particular it shows that the absolute ratio of our algorithm is 5/3, and this is optimal.…”

Section: Introductionmentioning

confidence: 58%

“…We use the construction by Dósa and Epstein [8] proving the lower bound 2 for two colors to get a lower bound 2.5 using three colors. We combine it with the hard instance that shows the lower bound 1.5 for zero-size items.…”

Section: Lower Bound On Competitiveness Of Any Online Algorithmmentioning

confidence: 99%

“…The second part of the instance is a slightly simplified construction by Dósa and Epstein [8]. Their idea goes as follows: The adversary sends the instance in phases, each starting with white and black small items.…”

Section: Lower Bound On Competitiveness Of Any Online Algorithmmentioning

confidence: 99%

“…Very recently and independent of us Dósa and Epstein [8] studied Colored Bin Packing. They improved the lower bound for online Black and White Bin Packing to 2 for deterministic algorithms, which holds for more colors as well.…”

Section: Introductionmentioning

confidence: 99%

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Online Colored Bin Packing

Böhm

Sgall

Veselý

2015

Approximation and Online Algorithms

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In the Colored Bin Packing problem a sequence of items of sizes up to 1 arrives to be packed into bins of unit capacity. Each item has one of c ≥ 2 colors and an additional constraint is that we cannot pack two items of the same color next to each other in the same bin. The objective is to minimize the number of bins.In the important special case when all items have size zero, we characterize the optimal value to be equal to color discrepancy. As our main result, we give an (asymptotically) 1.5-competitive algorithm which is optimal. 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. When the items have sizes of at most 1/d for a real d ≥ 2 the asymptotic competitive ratio is 1.5 + d/(d − 1). We also show that classical algorithms First Fit, Best Fit and Worst Fit are not constant competitive, which holds already for three colors and small items. In the case of two colors-the Black and White Bin Packing problem-we prove that all Any Fit algorithms have the absolute competitive ratio 3. When the items have sizes of at most 1/d for a real d ≥ 2 we show that the Worst Fit algorithm is absolutely (1 + d/(d − 1))-competitive.

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Section: Introductionmentioning

confidence: 58%

Section: Lower Bound On Competitiveness Of Any Online Algorithmmentioning

confidence: 99%

Section: Lower Bound On Competitiveness Of Any Online Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Online Colored Bin Packing

Böhm

Sgall

Veselý

2015

Approximation and Online Algorithms

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“…Most of the literature on colorful bin packing is about the online version of the problem. Competitive algorithms for the online colorful bin packing problem were presented in [11]. The special case black and white was considered in [2], while, the one where all items have size 0, was considered in [5].…”

Section: Introductionmentioning

confidence: 99%

On Colorful Bin Packing Games

Bilò

Cellinese

Melideo

et al. 2018

Lecture Notes in Computer Science

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We consider colorful bin packing games in which selfish players control a set of items which are to be packed into a minimum number of unit capacity bins. Each item has one of m ≥ 2 colors and cannot be packed next to an item of the same color. All bins have the same unitary cost which is shared among the items it contains, so that players are interested in selecting a bin of minimum shared cost. We adopt two standard cost sharing functions: the egalitarian cost function which equally shares the cost of a bin among the items it contains, and the proportional cost function which shares the cost of a bin among the items it contains proportionally to their sizes. 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 for computing a Nash equilibrium whose running time is polynomial under the egalitarian cost function and pseudo-polynomial for a constant number of colors under the proportional one. We also provide a complete characterization of the efficiency of Nash equilibria under both cost functions for general games, by showing that the prices of anarchy and stability are unbounded when m ≥ 3 while they are equal to 3 for black and white games, where m = 2. We finally focus on games with uniform sizes (i.e., all items have the same size) for which the two cost functions coincide. We show again a tight characterization of the efficiency of Nash equilibria and design an algorithm which returns Nash equilibria with best achievable performance.

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