# Covering Polygons with Rectangles

@inproceedings{Glck2017CoveringPW, title={Covering Polygons with Rectangles}, author={Roland Gl{\"u}ck}, booktitle={TAMC}, year={2017} }

A well-known and well-investigated family of hard optimization problems deals with nesting, i.e., the non-overlapping placing of polygons to be cut from a rectangle or the plane whilst minimizing the waste. Here we consider the in some sense inverse problem of a subsequent step in production technology: given a set of polygons in the plane and an axis-aligned rectangle (modeling a gripping device), we seek the minimum number of copies of the rectangle such that every polygon is completely…

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## References

SHOWING 1-10 OF 17 REFERENCES

### Rectangle covers revisited computationally

- Computer Science, MathematicsWEA
- 2005

An integer program is proposed which is the first general approach to obtain provably optimal solutions to this well-studied NP-hard problem of covering an orthogonal polygon with a minimum number of axis-parallel rectangles and a stronger lower bound on the optimum, namely, the cardinality of a fractional stable set is proposed.

### Optimal placement of convex polygons to maximize point containment

- Mathematics, Computer ScienceSODA '96
- 1996

### Covering Many or Few Points with Unit Disks

- Computer Science, MathematicsTheory of Computing Systems
- 2008

An algorithm is presented that computes, for any fixed ε>0, in O(nlog 2n) expected time a disk that is, with high probability, a (1+ε)-approximation to the optimal solution.

### Covering polygons is hard

- Mathematics, Computer Science[Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science
- 1988

It is shown that the following minimum cover problems are NP-hard, even for polygons without holes, and these results hold even if the polygons are required to be in general position.

### Approximation schemes for covering and packing problems in image processing and VLSI

- Computer ScienceJACM
- 1985

The unified technique that is introduced here, referred to as the shifting strategy, is applicable to numerous geometric covering and packing problems and how it varies with problem parameters is illustrated.

### Determining an upper bound for a class of rectangular packing problems

- Computer ScienceComput. Oper. Res.
- 1985

### Almost optimal set covers in finite VC-dimension

- Mathematics, Computer ScienceDiscret. Comput. Geom.
- 1995

We give a deterministic polynomial-time method for finding a set cover in a set system (X, ℛ) of dual VC-dimensiond such that the size of our cover is at most a factor ofO(d log(dc)) from the optimal…

### Covering rectilinear polygons with axis-parallel rectangles

- Computer Science, MathematicsSTOC '99
- 1999

This is the first polynomial time approximation algorithm for this problem with an O(\sqrt n) factor approximation factor for covering a rectilinear polygon with holes using axis-parallel rectangles.