# Polynomiality for Bin Packing with a Constant Number of Item Types

@inproceedings{Goemans2014PolynomialityFB, title={Polynomiality for Bin Packing with a Constant Number of Item Types}, author={Michel X. Goemans and Thomas Rothvoss}, booktitle={SODA}, year={2014} }

We consider the bin packing problem with d different item sizes s_i and item multiplicities a_i, where all numbers are given in binary encoding. This problem formulation is also known as the 1-dimensional cutting stock problem.
In this work, we provide an algorithm which, for constant d, solves bin packing in polynomial time. This was an open problem for all d >= 3.
In fact, for constant d our algorithm solves the following problem in polynomial time: given two d-dimensional polytopes P and Q…

## 72 Citations

Approximation and online algorithms for multidimensional bin packing: A survey

- Computer Science, BusinessComput. Sci. Rev.
- 2017

About the Structure of the Integer Cone and its Application to Bin Packing

- MathematicsSODA
- 2017

An algorithm for the bin packing problem with running time 2O(d) · enc(I)O(1) is obtained, parameterized by the number of vertices of the integer knapsack polytope |V|, which shows that the binpacking problem can be solved efficiently when the underlying integerknapsackpolytope has an easy structure.

Multidimensional Bin Packing and Other Related Problems : A Survey ∗

- Computer Science
- 2016

This survey considers several classical generalizations of bin packing problem such as geometric bin packing, vector bin packing and various other related problems such as vector scheduling, vector covering etc.

Solving Packing Problems with Few Small Items Using Rainbow Matchings

- Computer ScienceMFCS
- 2020

A colored matching problem is introduced to which fixed-parameter algorithms for vector versions of Bin Packing, Multiple Knapsack, and Bin Covering parameterized by $k are reduced, and the algorithms are randomized with one-sided error and run in time.

High Multiplicity Strip Packing Problem With Three Rectangle Types

- Business
- 2019

The two-dimensional strip packing problem (2D-SPP) involves packing a set R = {r1, ..., rn} of n rectangular items into a strip of width 1 and unbounded height, where each rectangular item ri has…

Knapsack and Subset Sum with Small Items

- Computer ScienceICALP
- 2021

These algorithms work for the more general problem variants with multiplicities, where each input item comes with a (binary encoded) multiplicity, which succinctly describes how many times the item appears in the instance.

Packing into designated and multipurpose bins: A theoretical study and application to the cold chain☆

- Business
- 2017

Parameterized complexity of Strip Packing and Minimum Volume Packing

- MathematicsTheor. Comput. Sci.
- 2017

High Multiplicity Scheduling on Uniform Machines in FPT-Time

- Computer ScienceArXiv
- 2022

This work considers scheduling on uniform machines where a job of size pj takes time pj/si on a machine of speed si and proposes an O(smax · p 3) max poly|I|) time algorithm, which is better than the running times of the algorithms known until today.

Beating the Harmonic lower bound for online bin packing

- Computer ScienceICALP
- 2016

This paper presents an online bin packing algorithm with asymptotic competitive ratio of 1.5813, the first improvement in fifteen years and reduces the gap to the lower bound by 15%.

## References

SHOWING 1-10 OF 31 REFERENCES

An asymptotically exact algorithm for the high-multiplicity bin packing problem

- Mathematics, Computer ScienceMath. Program.
- 2005

This work considers the high-multiplicity version of the bin packing problem, and shows that when C=2 the problem can be solved in time O( log D), where poly(·) is a polynomial function not depending on C.

An OPT + 1 Algorithm for the Cutting Stock Problem with Constant Number of Object Lengths

- Mathematics, Computer ScienceIPCO
- 2010

This paper considers the version of the problem in which the number d of different object types is constant and presents an algorithm that computes a solution using at most OPT+1 bins, where OPT is the value of an optimum solution.

An efficient approximation scheme for the one-dimensional bin-packing problem

- Computer Science23rd Annual Symposium on Foundations of Computer Science (sfcs 1982)
- 1982

It is proved that the LP relaxation of bin packing, which was solved efficiently in practice by Gilmore and Gomory, has membership in P, despite the fact that it has an astronomically large number of variables.

Strongly Polynomial Algorithms for the High Multiplicity Scheduling Problem

- Business, MathematicsOper. Res.
- 1991

This paper provides strongly polynomial algorithms for constructing optimal schedules with respect to several measures of efficiency (completion time, lateness, tardiness, the number of tardy jobs and their weighted counterparts) and identifies a new family of nxn transportation problems which are solvable in O(n log n) time by a simple greedy algorithm.

Polynomial algorithms for multiprocessor scheduling with a small number of job lengths

- BusinessSODA '97
- 1997

A polynomial-time algorithm is presented that produces a schedule that uses at most three different one-machine schedules, the minimum possible number, and is extended to the case of machine-dependent deadlines and to a multi-parametric case.

A Polynomial Algorithm for Multiprocessor Scheduling with Two Job Lengths

- BusinessMath. Oper. Res.
- 2001

A polynomial-time algorithm is presented that guarantees a schedule that uses the minimum possible number of different one-machine schedules, namely three, and is extended to the case of machine-dependent deadlines uniform parallel machines, to a multi-parametric case that contains the cases of unrelated parallel Machines, and to some related covering problems.

On the bin packing problem with a fixed number of object weights

- MathematicsEur. J. Oper. Res.
- 2007

Approximating Bin Packing within O(log OPT * Log Log OPT) Bins

- Computer Science2013 IEEE 54th Annual Symposium on Foundations of Computer Science
- 2013

It is shown that one can find a solution of cost OPT + O(log OPT * log log OPT) in polynomial time and this is achieved by rounding a fractional solution to the Gilmore-Gomory LP relaxation using the Entropy Method from discrepancy theory.

Minkowski's Convex Body Theorem and Integer Programming

- Mathematics, Computer ScienceMath. Oper. Res.
- 1987

An algorithm for solving Integer Programming problems whose running time depends on the number n of variables as nOn by reducing an n variable problem to 2n5i/2 problems in n-i variables for some i greater than zero chosen by the algorithm.

Reverse-Fit: A 2-Optimal Algorithm for Packing Rectangles

- MathematicsESA
- 1994

A ”level-oriented” algorithm, called ”Reverse-Fit”, for packing rectangles into a unit-width, infinite-height bin so as to minimize the total height of the packing.