# Framework for ER-Completeness of Two-Dimensional Packing Problems

@article{Abrahamsen2020FrameworkFE, title={Framework for ER-Completeness of Two-Dimensional Packing Problems}, author={Mikkel Abrahamsen and Tillmann Miltzow and Nadja Seiferth}, journal={2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)}, year={2020}, pages={1014-1021} }

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 in the resulting placement, they are pairwise interior-disjoint, and only motions of the allowed type can be used to move them…

## 26 Citations

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

- Mathematics
- 2020

It is shown that many natural two-dimensional packing problems are algorithmically equivalent to finding real roots of multivariate polynomials and to deciding whether a given system of polynomial equations and inequalities with integer coefficients has a real solution.

### Online Square Packing with Rotation

- Mathematics, Computer Science
- 2022

The online problem when item rotation is allowed is investigated, and a linear-time online algorithm is introduced that achieves an asymptotic competitive ratio of 2 .

### (Re)packing Equal Disks into Rectangle

- Computer ScienceICALP
- 2022

The problem of packing of equal disks (or circles) into a rectangle is a fundamental geometric problem. (By a packing here we mean an arrangement of disks in a rectangle without overlapping.) We…

### Packing Squares into a Disk with Optimal Worst-Case Density

- MathematicsSoCG
- 2021

This result settles the last (and arguably, most elusive) case of packing circular or square objects into a circular orsquare container and showcases the power of a general framework for computer-assisted proofs, based on interval arithmetic.

### Geometric Embeddability of Complexes is ∃R-complete

- MathematicsArXiv
- 2021

We show that the decision problem of determining whether a given (abstract simplicial) k-complex has a geometric embedding in R is complete for the Existential Theory of the Reals for all d ≥ 3 and k…

### Geometric Embeddability of Complexes is $\exists \mathbb R$-complete

- Mathematics
- 2021

We show that the decision problem of determining whether a given (abstract simplicial) k-complex has a geometric embedding in R is complete for the Existential Theory of the Reals for all d ≥ 3 and k…

### A Framework for Robust Realistic Geometric Computations

- Computer Science, MathematicsArXiv
- 2019

It is proved that suitable algorithms can (under smoothed analysis) be robustly executed with expected logarithmic bit-precision and concluded with a real RAM analogue to the Cook-Levin Theorem, which gives an easy proof of ER-membership.

### Smoothing the gap between NP and ER

- Computer Science, Mathematics2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
- 2020

A real RAM analogue to the Cook-Levin theorem is proved which shows that ER membership is equivalent to having a verification algorithm that runs in polynomial-time on a real RAM, which gives an easy proof of ER-membership.

### 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.

### Complexity of the Boundary-Guarding Art Gallery Problem

- Mathematics
- 2022

We resolve the complexity of the boundary-guarding variant of the art gallery problem, showing that it is ∃ R -complete, meaning that it is equivalent under polynomial time reductions to deciding…

## References

SHOWING 1-10 OF 55 REFERENCES

### 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.

### Hardness of Approximation for Strip Packing

- MathematicsTOCT
- 2017

This article proves that it is NP-hard to approximate strip packing within a factor better than 12/11, even when restricted to polynomially bounded input data, and shows that the strip packing problem admits no quasi-polynomial time approximation scheme, unless NP} ⊑ DTIME(2polylog (n)).

### 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.

### 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.

### 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.

### Some provably hard crossing number problems

- MathematicsSCG '90
- 1990

It is shown that any given arrangement can be forced to occur in every minimum-crossing drawing of an appropriate graph, and that there exists no polynomial-time algorithm for producing a straight-line drawing of a graph, with minimum number of crossings from among all such drawings.

### Approximating Geometric Knapsack via L-Packings

- Computer Science, Mathematics2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
- 2017

This paper breaks the 2 approximation barrier, achieving a polynomialtime 17/9 + ε and aims to find a (non-overlapping) packing of a maximum profit subset of items inside the knapsack.