WADS 2011: Algorithms and Data Structures pp 231-242

# Beyond Triangulation: Covering Polygons with Triangles

• Tobias Christ
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6844)

## Abstract

We consider the triangle cover problem. Given a polygon P, cover it with a minimum number of triangles contained in P. This is a generalization of the well-known polygon triangulation problem. Another way to look at it is as a restriction of the convex cover problem, in which a polygon has to be covered with a minimum number of convex pieces. Answering a question stated in the Handbook of Discrete and Computational Geometry, we show that the convex cover problem without Steiner points is NP-hard. We present a reduction that also implies NP-hardness of the triangle cover problem and which in a second step allows to get rid of Steiner points. For the problem where only the boundary of the polygon has to be covered, we also show that it is contained in NP and thus NP-complete and give an efficient factor 2 approximation algorithm.

## Keywords

Cover Problem Steiner Point Simple Polygon Polygonal Region Main Switch
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Keil, J.M.: Polygon decomposition. In: Handbook of Computational Geometry, pp. 491–518. North-Holland, Amsterdam (2000)
2. 2.
Culberson, J.C., Reckhow, R.A.: Covering polygons is hard. J. Algorithms 17(1), 2–44 (1994)
3. 3.
O’Rourke, J., Suri, S.: Polygons. In: Goodman, J.E., O’Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, pp. 583–606. CRC Press, LLC, Boca Raton, FL (2004)Google Scholar
4. 4.
Mulzer, W., Rote, G.: Minimum weight triangulation is NP-hard. In: Proc. 22nd Annu. ACM Sympos. Comput. Geom., pp. 1–10 (2006)Google Scholar
5. 5.
O’Rourke, J.: Art gallery theorems and algorithms. International Series of Monographs on Computer Science. The Clarendon Press Oxford University Press, New York (1987)
6. 6.
Ghosh, S.K.: Approximation algorithms for art gallery problems in polygons. Discrete Appl. Math. 158(6), 718–722 (2010)
7. 7.
Eidenbenz, S.J., Widmayer, P.: An approximation algorithm for minimum convex cover with logarithmic performance guarantee. SIAM J. Comput. 32(3), 654–670 (2003) (electronic)
8. 8.
O’Rourke, J., Supowit, K.J.: Some NP-hard polygon decomposition problems. IEEE Trans. Inform. Theory IT-30, 181–190 (1983)
9. 9.
Chazelle, B., Dobkin, D.P.: Optimal convex decompositions. In: Computational geometry. Mach. Intelligence Pattern Recogn., vol. 2, pp. 63–133. North-Holland, Amsterdam (1985)
10. 10.
Chen, C., Chang, R.: On the minimality of polygon triangulation. BIT 30(4), 570–582 (1990)
11. 11.
Lingas, A.: The power of non-rectilinear holes. In: Nielsen, M., Schmidt, E.M. (eds.) ICALP 1982. LNCS, vol. 140, pp. 369–383. Springer, Heidelberg (1982)
12. 12.
Asano, T., Asano, T., Pinter, R.Y.: Polygon triangulation: Efficiency and minimality. J. Algorithms 7(2), 221–231 (1986)
13. 13.
Keil, M., Snoeyink, J.: On the time bound for convex decomposition of simple polygons. Internat. J. Comput. Geom. Appl. 12(3), 181–192 (2002)
14. 14.
Chazelle, B.: Triangulating a simple polygon in linear time. Discrete Comput. Geom. 6(5), 485–524 (1991)
15. 15.
Arora, S.: Exploring complexity through reductions. In: Computational complexity theory. IAS/Park City Math. Ser., vol. 10, pp. 101–126. Amer. Math. Soc., Providence (2004)Google Scholar
16. 16.
Suri, S., O’Rourke, J.: Worst-case optimal algorithms for constructing visibility polygons with holes. In: Proc. 2nd Annu. ACM Sympos. Comput. Geom., pp. 14–23 (1986)Google Scholar
17. 17.
Chazelle, B., Edelsbrunner, H., Grigni, M., et al.: Ray shooting in polygons using geodesic triangulations. Algorithmica 12(1), 54–68 (1994)
18. 18.
Mucha, M., Sankowski, P.: Maximum matchings via gaussian elimination. In: FOCS, pp. 248–255. IEEE Computer Society, Los Alamitos (2004)Google Scholar