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Online bin packing with (1,1) and (2,\(R\)) bins

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Abstract

We study a variant of online bin packing problem, in which there are two types of bins: \((1,1)\) and \((2,R)\), i.e., unit size bin with cost 1 and size 2 bin with cost \(R > 1\), the objective is to minimize the total cost occurred when all the items are packed into the two types of bins. When \(R > 3\), the offline version of this problem is equivalent to the classical bin packing problem. In this paper, we focus on the case \( R \le 3\), and propose online algorithms and obtain lower bounds for the problem.

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Acknowledgments

Partially supported by NSFC(11101065), LJQ2012003, RGC(HKU716412E) and “the Fundamental Research Funds for the Central Universities”

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Authors and Affiliations

  1. Graduate School of Informatics, Kyoto University, Kyoto, Japan

    Jing Chen & Kazuo Iwama

  2. Software School, Dalian University of Technology, Dalian, People’s Republic of China

    Xin Han

  3. Department of Computer Science, The University of Hong Kong, Pokfulam, Hong Kong

    Hing-Fung Ting

Authors
  1. Jing Chen
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  2. Xin Han
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  3. Kazuo Iwama
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  4. Hing-Fung Ting
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Corresponding author

Correspondence to Xin Han.

Appendix

Appendix

In the following, we use \([p_1, p_2, p_3, p_4]\) to denote a dominant pattern of size 2 bin, where the number of items with size \(s_i\) is \(p_i\). And we use \((p_1, p_2, p_3, p_4)\) to denote a dominant pattern of a unit bin, where the number of items with size \(s_i\) is \(p_i\).

1.1 Dominant patterns of Greedy sequence

Dominant patterns for size two bin: [84,0,0,0], [78,1,0,0], [72,2,0,0], [66,3,0,0], [60,4,0,0], [54,5,0,0], [48,6,0,0], [42,7,0,0], [36,8,0,0], [30,9,0,0], [24,10,0,0], [18,11,0,0], [12,12,0,0], [6,13,0,0], [70,0,1,0], [64,1,1,0], [58,2,1,0], [52,3,1,0], [46,4,1,0], [40,5,1,0], [34,6,1,0], [28,7,1,0], [22,8,1,0], [16,9,1,0], [10,10,1,0], [4,11,1,0], [56, 0,2,0], [50,1,2,0], [44,2,2,0], [38,3,2,0], [32,4,2,0], [26,5,2,0], [20,6,2,0], [14,7,2,0], [8,8,2,0], [1,9,2,0], [42,0,3,0], [36,1,3,0], [30,2,3,0], [24,3,3,0, ], [18,4,3,0], [12,5,3,0], [6,6,3,0], [28,0,4,0], [22,1,4,0], [16,2,4,0], [10,3,4,0], [4,4,4,0], [14,0,5,0], [8,1,5,0], [1,2,5,0], [63,0,0,1], [57,1,0,1], [51,2, 0,1], [45,3,0,1], [39,4,0,1], [33,5,0,1], [27,6,0,1], [21,7,0,1], [15,8,0,1], [9,9,0,1], [2,10,0,1], [49,0,1,1], [43,1,1,1], [37,2,1,1], [31,3,1,1], [25,4,1,1], [19,5,1,1], [13,6,1,1], [7,7,1,1], [0,8,1,1], [35,0,2,1], [29,1,2,1], [23,2,2,1], [17,3,2,1], [11,4,2,1], [5,5,2,1], [21,0,3,1], [15,1,3,1], [9,2,3,1], [3,3,3,1], [7,0,4,1], [0,1,4,1], [42,0,0,2], [36,1,0,2], [30,2,0,2], [24,3,0,2], [18,4,0,2], [12,5,0,2], [6,6,0,2], [28,0,1,2], [22,1,1,2], [16,2,1,2], [10,3,1,2], [4,4, 1,2], [14,0,2,2], [8,1,2,2], [2,2,2,2], [21,0,0,3], [15,1,0,3], [9,2,0,3], [3,3,0,3], [7,0,1,3], [0,1,1,3], [0,13,0,0], [0,11,1,0], [0,9,2,0], [0,6,3,0], [0,4,4, 0], [0,2,5,0], [0,10,0,1], [0,8,1,1], [0,5,2,1], [0,3,3,1], [0,1,4,1], [0,6,0,2], [0,4,1,2], [0,2,2,2], [0,3,0,3], [0,1,1,3], [0,0,5,0], [0,0,4,1], [0,0,2,2], [0,0,1,3], [0,0,0,3]

Dominant patterns for a unit bin:

(42,0,0,0), (36,1,0,0), (30,2,0,0), (24,3,0,0), (18,4,0,0), (12,5,0,0), (6,6,0,0), (28,0,1,0), (22,1,1,0), (16,2,1,0), (10,3,1,0), (4,4,1,0), (14,0,2,0), (8,1, 2,0), (2,2,2,0), (21,0,0,1), (15,1,0,1), (9,2,0,1), (3,3,0,1), (7,0,1,1), (1,1,1,1), (0,6,0,0), (0,4,1,0), (0,2,2,0), (0,3,0,1), (0,1,1,1), (0,0,2,0), (0,0,1,1), (0,0,0,1)

1.2 Dominant patterns of Parametric sequence

Dominant patterns for size two bin: [156,0,0,0], [144,1,0,0], [132,2,0,0], [120,3,0,0], [108,4,0,0], [96,5,0,0], [84,6,0,0], [72,7,0,0], [60,8,0,0], [48,9,0,0], [36,10,0,0], [24,11,0,0], [12,12,0, 0], [117,0,1,0], [105,1,1,0], [93,2,1,0], [81,3,1,0], [69,4,1,0], [57,5,1,0], [45,6,1,0], [33,7,1,0], [21,8,1,0], [9,9,1,0], [78,0,2,0], [66,1,2,0], [54,2,2,0], [42,3,2,0], [30,4,2,0], [18,5,2,0], [6,6,2,0], [39,0,3,0], [27,1,3,0], [15,2,3,0], [3,3,3,0], [104,0,0,1], [92,1,0,1], [80,2,0,1], [68,3,0,1], [56,4,0,1], [44,5, 0,1], [32,6,0,1], [20,7,0,1], [8,8,0,1], [65,0,1,1], [53,1,1,1], [41,2,1,1], [29,3,1,1], [17,4,1,1], [5,5,1,1], [26,0,2,1], [14,1,2,1], [2,2,2,1], [52,0,0,2], [40,1,0,2], [28,2,0,2], [16,3,0,2], [4,4,0,2], [13,0,1,2], [0,1,1,2], [0,12,0,0], [0,9,1,0], [0,6,2,0], [0,3,3,0], [0,8,0,1], [0,5,1,1], [0,2,2,1], [0,4,0,2], [0,1,1,2], [0,0,3,0], [0,0,2,1], [0,0,1,2], [0,0,0,2]

Dominant patterns for a unit bin:

(78,0,0,0), (66,1,0,0), (54,2,0,0), (42,3,0,0), (30,4,0,0), (18,5,0,0), (6,6,0,0), (39,0,1,0), (27,1,1,0), (15,2,1,0), (3,3,1,0), (26,0,0,1), (14,1,0,1), (2,2, 0,1), (0,6,0,0), (0,3,1,0), (0,2,0,1), (0,0,1,0), (0,0,0,1)

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Chen, J., Han, X., Iwama, K. et al. Online bin packing with (1,1) and (2,\(R\)) bins. J Comb Optim 30, 276–298 (2015). https://doi.org/10.1007/s10878-014-9749-6

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  • DOI: https://doi.org/10.1007/s10878-014-9749-6

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