### Abstract

We define and study a variant of bin packing called unrestricted black and white bin packing. Similarly to standard bin packing, a set of items of sizes in [0, 1] are to be partitioned into subsets of total size at most 1, called bins. Items are of two types, called black and white, and the item types must alternate in each bin, that is, two items of the same type cannot be assigned consecutively into a bin. Thus, a subset of items of total size at most 1 can form a valid bin if and only if the absolute value of the difference between the numbers of black items and white items in the subset is at most 1. We study this problem both with respect to the absolute and the asymptotic approximation ratios. We design a fast heuristic whose absolute approximation ratio is 2. We also design an APTAS and modify it into an AFPTAS. The APTAS can be used as an algorithm of absolute approximation ratio 32, which is the best possible absolute approximation ratio for the problem unless P. =. NP.

Original language | English |
---|---|

Pages (from-to) | 92-101 |

Number of pages | 10 |

Journal | Theoretical Computer Science |

Volume | 596 |

DOIs | |

Publication status | Published - Sep 6 2015 |

### Fingerprint

### Keywords

- Approximation schemes
- Bin packing
- First Fit
- Knapsack
- Next Fit

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Theoretical Computer Science*,

*596*, 92-101. https://doi.org/10.1016/j.tcs.2015.06.045

**Offline black and white bin packing.** / Balogh, János; Békési, József; Dósa, G.; Epstein, Leah; Kellerer, Hans; Levin, Asaf; Tuza, Z.

Research output: Contribution to journal › Article

*Theoretical Computer Science*, vol. 596, pp. 92-101. https://doi.org/10.1016/j.tcs.2015.06.045

}

TY - JOUR

T1 - Offline black and white bin packing

AU - Balogh, János

AU - Békési, József

AU - Dósa, G.

AU - Epstein, Leah

AU - Kellerer, Hans

AU - Levin, Asaf

AU - Tuza, Z.

PY - 2015/9/6

Y1 - 2015/9/6

N2 - We define and study a variant of bin packing called unrestricted black and white bin packing. Similarly to standard bin packing, a set of items of sizes in [0, 1] are to be partitioned into subsets of total size at most 1, called bins. Items are of two types, called black and white, and the item types must alternate in each bin, that is, two items of the same type cannot be assigned consecutively into a bin. Thus, a subset of items of total size at most 1 can form a valid bin if and only if the absolute value of the difference between the numbers of black items and white items in the subset is at most 1. We study this problem both with respect to the absolute and the asymptotic approximation ratios. We design a fast heuristic whose absolute approximation ratio is 2. We also design an APTAS and modify it into an AFPTAS. The APTAS can be used as an algorithm of absolute approximation ratio 32, which is the best possible absolute approximation ratio for the problem unless P. =. NP.

AB - We define and study a variant of bin packing called unrestricted black and white bin packing. Similarly to standard bin packing, a set of items of sizes in [0, 1] are to be partitioned into subsets of total size at most 1, called bins. Items are of two types, called black and white, and the item types must alternate in each bin, that is, two items of the same type cannot be assigned consecutively into a bin. Thus, a subset of items of total size at most 1 can form a valid bin if and only if the absolute value of the difference between the numbers of black items and white items in the subset is at most 1. We study this problem both with respect to the absolute and the asymptotic approximation ratios. We design a fast heuristic whose absolute approximation ratio is 2. We also design an APTAS and modify it into an AFPTAS. The APTAS can be used as an algorithm of absolute approximation ratio 32, which is the best possible absolute approximation ratio for the problem unless P. =. NP.

KW - Approximation schemes

KW - Bin packing

KW - First Fit

KW - Knapsack

KW - Next Fit

UR - http://www.scopus.com/inward/record.url?scp=84937702231&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84937702231&partnerID=8YFLogxK

U2 - 10.1016/j.tcs.2015.06.045

DO - 10.1016/j.tcs.2015.06.045

M3 - Article

VL - 596

SP - 92

EP - 101

JO - Theoretical Computer Science

JF - Theoretical Computer Science

SN - 0304-3975

ER -