# Approximating the Maximum Overlap of Polygons under Translation

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## Abstract

Let \(P\) and \(Q\) be two simple polygons in the plane of total complexity *n*, each of which can be decomposed into at most *k* convex parts. We present a \((1-\varepsilon )\)-approximation algorithm, for finding the translation of \(Q\), which maximizes its area of overlap with \(P\). Our algorithm runs in \(O\left( {c n}\right) \) time, where *c* is a constant that depends only on *k* and \(\varepsilon \). This suggests that for polygons that are “close” to being convex, the problem can be solved (approximately), in near linear time.

## Keywords

Approximation algorithms Computational geometry Polygon overlap## Notes

### Acknowledgments

The authors would like to thank the anonymous referees for their insightful comments. In particular, the improved construction of Lemma 12 was suggested by an anonymous referee.

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