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

@article{Hochbaum1985ApproximationSF, title={Approximation schemes for covering and packing problems in image processing and VLSI}, author={D. Hochbaum and W. Maass}, journal={J. ACM}, year={1985}, volume={32}, pages={130-136} }

A unified and powerful approach is presented for devising polynomial approximation schemes for many strongly NP-complete problems. Such schemes consist of families of approximation algorithms for each desired performance bound on the relative error ε > &Ogr;, with running time that is polynomial when ε is fixed. Though the polynomiality of these algorithms depends on the degree of approximation ε being fixed, they cannot be improved, owing to a negative result stating that there are no fully… Expand

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#### References

SHOWING 1-10 OF 11 REFERENCES

Approximation algorithms for NP-complete problems on planar graphs

- Mathematics, Computer Science
- 24th Annual Symposium on Foundations of Computer Science (sfcs 1983)
- 1983

A general technique that can be used to obtain approximation algorithms for various NP-complete problems on planar graphs, which includes maximum independent set, maximum tile salvage, partition into triangles, maximum H-matching, minimum vertex cover, minimum dominating set, and minimum edge dominating set. Expand

Optimal Packing and Covering in the Plane are NP-Complete

- Computer Science
- Inf. Process. Lett.
- 1981

This paper proves that even severely restricted instances of packing and covering problems remain NP-hard in two or more dimensions, and helps to fill the gap by showing that some very constrained intersection graph problems in two dimensions are not very constrained. Expand

Optimal Tile Salvage.

- Mathematics
- 1981

Abstract : A map is a tiling of a finite region of the plane with unit squares such that some tiles have been removed. The optimal x x y-Tile Salvage Problem is: Given a map, find the maximum number… Expand

The NP-Completeness Column: An Ongoing Guide

- Mathematics, Computer Science
- J. Algorithms
- 1986

This is the fourteenth edition of a quarterly column that provides continuing coverage of new developments in the theory of NP-completeness, and readers who have results they would like mentioned (NP-hardness, PSPACE- hardness, polynomialtime-solvability, etc.), or open problems they wouldlike publicized, should send them to David S. Johnson. Expand

Computers and Intractability: A Guide to the Theory of NP-Completeness

- Computer Science, Mathematics
- 1978

Horn formulae play a prominent role in artificial intelligence and logic programming. In this paper we investigate the problem of optimal compression of propositional Horn production rule knowledge… Expand

Fast approximation schemes for geometric covering and packing problems in robotics and VLSI

- Unpublished manuscript,
- 1983

Fast approximation schemes for geometric covering and packing problems in robotics and VLSI. Unpublished manuscript

- Fast approximation schemes for geometric covering and packing problems in robotics and VLSI. Unpublished manuscript
- 1983

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