# Online Colored Bin Packing

@article{Bhm2014OnlineCB, title={Online Colored Bin Packing}, author={Martin B{\"o}hm and Jir{\'i} Sgall and Pavel Vesel{\'y}}, journal={ArXiv}, year={2014}, volume={abs/1404.5548} }

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 \(c\ge 2\) 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.

#### Supplemental Presentations

#### Figures and Topics from this paper

#### 17 Citations

Bin Packing with Multiple Colors

- Mathematics, Computer Science
- ArXiv
- 2015

This work solves the Colored Bin Packing problem for the case where there are two or more colors when the items have zero weight and when theItems have unit weight. Expand

A Two-Pass Algorithm for Unordered Colored Bin Packing

- Computer Science, Mathematics
- DOOR
- 2016

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… Expand

Colored Bin Packing: Online Algorithms and Lower Bounds

- Computer Science, Mathematics
- Algorithmica
- 2016

This work gives an (asymptotically) 1.5-competitive algorithm for Colored Bin Packing problem and shows that classical algorithms First Fit, Best Fit and Worst Fit are not constant competitive, which holds already for three colors and small items. Expand

VNS matheuristic for a bin packing problem with a color constraint

- Mathematics, Computer Science
- Electron. Notes Discret. Math.
- 2017

This work studies a new variant of the bin packing problem that wants to pack all items into the minimal number of bins and applies the column generation technique based on the VNS matheuristic for the pricing problem. Expand

Black and White Bin Packing Revisited

- Mathematics, Computer Science
- COCOA
- 2015

The first 'better than 3' competitive algorithm is given for the black and white bin packing problem, where in addition to a size, each item also has a color black or white, and in each bin the colors of items must alternate. Expand

Binary Decision Diagrams for Bin Packing with Minimum Color Fragmentation

- Computer Science
- CPAIOR
- 2019

This work introduces the BPMCF and presents a decomposition strategy to solve the problem, where the assignment of items to bins is formulated as a binary decision diagram and an optimal integrated solutions is identified through a mixed-integer linear programming model. Expand

A Core Heuristic and the Branch-and-Price Method for a Bin Packing Problem with a Color Constraint

- Computer Science
- 2018

This work designs the core heuristic based on the column generation approach for the large-scale formulation of bin packing problem with a color constraint and uses it in the exact branch-and-price method. Expand

Bin packing with directed stackability conflicts

- Mathematics
- 2015

Abstract 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… Expand

Online packing of arbitrary sized items into designated and multipurpose bins

- Computer Science, Mathematics
- Eur. J. Oper. Res.
- 2019

The first-fit method is extended to the problem setting and proves an absolute competitive ratio bound that is a function of the bin costs, and an upper bound of 1.750 is established on the worst-case asymptotic (as the number of items grows large) competitive ratio. Expand

Offline black and white bin packing

- Mathematics, Computer Science
- Theor. Comput. Sci.
- 2015

The APTAS can be used as an algorithm of absolute approximation ratio 3 2 , which is the best possibleabsolute approximation ratio for the problem unless P = NP, and is designed and modified into an AFPTAS. Expand

#### References

SHOWING 1-10 OF 24 REFERENCES

Bin packing can be solved within 1 + ε in linear time

- Mathematics
- 1981

For any listL ofn numbers in (0, 1) letL* denote the minimum number of unit capacity bins needed to pack the elements ofL. We prove that, for every positive ε, there exists anO(n)-time algorithmS… Expand

Online bin packing with cardinality constraints revisited

- Computer Science, Mathematics
- ArXiv
- 2014

It is shown that First Fit has an absolute competitive ratio of 2 for k=4, but not for larger values of k, and a complete analysis of its asymptotic competitive ratio is presented, and it is proved that tight bounds of 2 are proved for any k \geq 4. Expand

Algorithms for on-line bin-packing problems with cardinality constraints

- Mathematics, Computer Science
- Discret. Appl. Math.
- 2004

It is shown that, for increasing values of k, the bound on the asymptotic worst-case performance ratio of the first algorithm tends towards 2 and that the second algorithm has a ratio of 2.7. Expand

Online Results for Black and White Bin Packing

- Mathematics, Computer Science
- Theory of Computing Systems
- 2014

A new variant where 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, which generalizes the standard online bin packing problem. Expand

Online Bin Packing with Cardinality Constraints

- Mathematics, Computer Science
- SIAM J. Discret. Math.
- 2006

The exact best competitive ratio for bounded space online algorithms for every value of $k$ is established, which tends to $\Pi_\infty+1\approx 2.69103$ for large $k$. Expand

Black and White Bin Packing

- Mathematics, Computer Science
- WAOA
- 2012

The competitiveness of some classical algorithms is studied, the universal lower bound for all online algorithms is proved, and an online algorithm is designed which is 3-competitive in the absolute sense. Expand

Colorful Bin Packing

- Mathematics, Computer Science
- SWAT
- 2014

A new algorithm for colorful bin packing is designed and analyzed, whose absolute competitive ratio is 4, and a new lower bound of 2 on the asymptotic competitive ratio of any algorithm is found, that is valid even for black and white bin packing. Expand

Improved Lower Bounds for the Online Bin Packing Problem with Cardinality Constraints

- Computer Science, Mathematics
- COCOON
- 2013

The k-item bin packing problem is one of the variants introduced by Krause et al. in Journal of the ACM 22(4) and imposes a cardinality constraint that the number of items packed into each bin must be at most k. Expand

First Fit bin packing: A tight analysis

- Mathematics, Computer Science
- STACS
- 2013

It is proved that if the optimum needs OPT bins, FirstFit always uses at most \lfloor 1.7 OPT \rfloor bins, and matching lower bounds for a majority of values of OPT are shown, which were previously known only for finitely many small values ofOpt. Expand

On the worst-case performance of the NkF bin-packing heuristic

- Mathematics, Computer Science
- Acta Cybern.
- 1989

The paper is interested in the worst-case behaviour of the Next-k Fit' (NkF) algorithm, and an upper and a lower bound were given in Johnson's paper. Expand