# A (2+\epsilon)-Approximation Algorithm for Maximum Independent Set of Rectangles

@inproceedings{Galvez2021AA, title={A (2+\epsilon)-Approximation Algorithm for Maximum Independent Set of Rectangles}, author={Waldo G'alvez and Arindam Khan and Mathieu Mari and Tobias Momke and Madhusudhan Reddy and Andreas Wiese}, year={2021} }

We study the Maximum Independent Set of Rectangles (MISR) problem, where we are given a set of axis-parallel rectangles in the plane and the goal is to select a subset of non-overlapping rectangles of maximum cardinality. In a recent breakthrough, Mitchell [45] obtained the first constant-factor approximation algorithm for MISR. His algorithm achieves an approximation ratio of 10 and it is based on a dynamic program that intuitively recursively partitions the input plane into special polygons…

## Figures from this paper

## References

SHOWING 1-10 OF 49 REFERENCES

### Approximating Maximum Independent Set for Rectangles in the Plane

- Computer Science, Mathematics2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

A polynomial-time constant-factor approximation algorithm for maximum independent set for (axis-aligned) rectangles in the plane is given, based on a new form of recursive partitioning inThe plane, in which faces that are constant-complexity and orthogonally convex are recursively partitioned into a constant number of such faces.

### On Guillotine Separability of Squares and Rectangles

- MathematicsAPPROX-RANDOM
- 2020

It is shown that O(1)-fraction of rectangles, even in the weighted case, can be recovered for many special cases of Rectangles, e.g. fat (bounded width/height), δ-large (large in one of the dimensions), etc.

### Coloring and Maximum Weight Independent Set of Rectangles

- Mathematics, Computer ScienceSODA
- 2021

The first asymptotic improvement over this six-decade-old bound is presented, proving that every intersection graph of axis-parallel rectangles in the plane admits an $O(\omega^2)$-coloring and presenting a polynomial-time algorithm that finds such a coloring.

### Polynomial-Time Approximation Schemes for Geometric Intersection Graphs

- Mathematics, Computer ScienceSIAM J. Comput.
- 2005

These are the first known PTASs for $\mathcal{NP}$-hard optimization problems on disk graphs based on a novel recursive subdivision of the plane that allows applying a shifting strategy on different levels simultaneously, so that a dynamic programming approach becomes feasible.

### Approximation algorithms for maximum independent set of pseudo-disks

- Computer ScienceSCG '09
- 2009

This work presents approximation algorithms for maximum independent set of pseudo-disks in the plane, both in the weighted and unweighted cases, and suggests a novel rounding scheme based on an LP relaxation of the problem that leads to a constant-factor approximation.

### On guillotine cutting sequences. In Approximation, Randomization, and Combinatorial Optimization

- Algorithms and Techniques (APPROX/RANDOM),
- 2015

### Improved Approximation Algorithms for 2-Dimensional Knapsack: Packing into Multiple L-Shapes, Spirals, and More

- Computer Science, MathematicsSoCG
- 2021

This paper gives (4/3 + ε)-approximation algorithms in polynomial time for both 2-Dimensional Knapsack problem and presents an algorithm that computes the essentially optimal structured packing into these regions.

### On Guillotine Separable Packings for the Two-dimensional Geometric Knapsack Problem

- Mathematics, Computer ScienceSoCG
- 2021

A structural lemma is shown which shows that any guillotine packing can be converted into another structured guillsotine packing with almost the same profit, which might be useful for other settings where these constraints are imposed.

### A Tight (3/2+ε) Approximation for Skewed Strip Packing

- Computer ScienceAPPROX-RANDOM
- 2020

This work considers the complementary case where all the rectangles are skewed and provides an (almost) tight (3/2 + ε)-approximation algorithm for Strip Packing.

### Breaking the Barrier of 2 for the Storage Allocation Problem

- Computer Science, MathematicsICALP
- 2020

This paper breaks the barrier of 2 for the Storage Allocation Problem (SAP) which is a natural intermediate problem between 2DKP and UFP and presents a polynomial time (63/32) < 1.969-approximation algorithm for the case of uniform edge capacities and a quasi-polynomial time (1.997)-approximating algorithm for non-uniform quasi- polynomially bounded edge capacities.