# Framework for $\exists \mathbb{R}$-Completeness of Two-Dimensional Packing Problems

@article{Abrahamsen2020FrameworkF, title={Framework for \$\exists \mathbb\{R\}\$-Completeness of Two-Dimensional Packing Problems}, author={Mikkel Abrahamsen and Tillmann Miltzow and Nadja Seiferth}, journal={arXiv: Computational Geometry}, year={2020} }

We show that many natural two-dimensional packing problems are algorithmically equivalent to finding real roots of multivariate polynomials. A two-dimensional packing problem is defined by the type of pieces, containers, and motions that are allowed. The aim is to decide if a given set of pieces can be placed inside a given container. The pieces must be placed so that they are pairwise interior-disjoint, and only motions of the allowed type can be used to move them there. We establish a…

## References

SHOWING 1-10 OF 77 REFERENCES

### Covering Polygons is Even Harder

- Mathematics, Computer Science2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

It is proved that assuming the widespread belief that NP-hard MCC is not in N P, and the problem is thus $\exists \mathbb{R}$-complete, that many natural approaches to finding small covers are bound to give suboptimal solutions in some cases.

### Computational Aspects of Packing Problems

- BusinessBull. EATCS
- 2016

A survey on results achieved about computability and complexity of packing problems, about approximation algorithms, and about very natural packing problems whose computational complexity is unknown.

### Closing the gap for pseudo-polynomial strip packing

- Computer ScienceESA
- 2019

This paper presents an algorithm with approximation ratio $(5/4 + \varepsilon)$ for Strip Packing and uses a structural result which states that each optimal solution can be transformed such that it has one of a polynomial number of different forms.

### On the Two-Dimensional Knapsack Problem for Convex Polygons

- Computer Science, MathematicsICALP
- 2020

These are the first results for two-dimensional geometric knapsack in which the input objects are more general than axis-parallel rectangles or circles and in whichThe input polygons can be rotated by arbitrary angles.

### A quasi-PTAS for the Two-Dimensional Geometric Knapsack Problem

- Computer ScienceSODA
- 2015

The key technical contribution is to show the existence of a partition for the knapsack into a small number of rectangular boxes, so powerful that it does not even need to round the sizes of the items, which is a canonical step in algorithms for geometricknapsack, geometric bin-packing, etc.

### Optimal Curve Straightening is $\exists\mathbb{R}$-Complete.

- Mathematics
- 2019

We prove that the following problem has the same computational complexity as the existential theory of the reals: Given a generic self-intersecting closed curve $\gamma$ in the plane and an integer…

### Realizability of Graphs and Linkages

- Mathematics
- 2013

We show that deciding whether a graph with given edge lengths can be realized by a straight-line drawing has the same complexity as deciding the truth of sentences in the existential theory of the…

### On approximating strip packing with a better ratio than 3/2

- Computer ScienceSODA
- 2016

This paper presents a (1.4 + e)-approximation algorithm with pseudo-polynomial running time that implies that for polynomially bounded input data the problem can be approximated with a strictly better ratio than for exponential input which is a very rare phenomenon in combinatorial optimization.