# Covering and Packing of Triangles Intersecting a Straight Line

• Supantha Pandit
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11394)

## Abstract

We study the four geometric optimization problems: , , , and with (a triangle is a right-triangle whose base is parallel to the x-axis, perpendicular is parallel to the y-axis, and the slope of the hypotenuse is $$-1$$). The input triangles are constrained to be intersecting a . The straight line can either be a or an line (a line whose slope is $$-1$$). A right-triangle is said to be a , if the length of both its base and perpendicular is $$\lambda$$. For $$1$$-right-triangles where the triangles intersect an inclined line, we prove that the set cover and hitting set problems are $$\mathsf {NP}$$-hard, whereas the piercing set and independent set problems are in $$\mathsf {P}$$. The same results hold for $$1$$-right-triangles where the triangles are intersecting a horizontal line instead of an inclined line. We prove that the piercing set and independent set problems with right-triangles intersecting an inclined line are $$\mathsf {NP}$$-hard. Finally, we give an $$n^{O(\lceil \log c\rceil +1)}$$ time exact algorithm for the independent set problem with $$\lambda$$-right-triangles intersecting a straight line such that $$\lambda$$ takes more than one value from [1, c], for some integer c. We also present $$O(n^2)$$ time dynamic programming algorithms for the independent set problem with $$1$$-right-triangles where the triangles intersect a horizontal line and an inclined line.

## Keywords

Set cover Hitting set Piercing set Independent set Horizontal line Inclined line Diagonal line $$\mathsf {NP}$$-hard Right triangles Dynamic programming.

## References

1. 1.
Catanzaro, D., et al.: Max point-tolerance graphs. Discret. Appl. Math. 216, 84–97 (2017)
2. 2.
Chepoi, V., Felsner, S.: Approximating hitting sets of axis-parallel rectangles intersecting a monotone curve. Comput. Geom. 46(9), 1036–1041 (2013)
3. 3.
Correa, J., Feuilloley, L., Pérez-Lantero, P., Soto, J.A.: Independent and hitting sets of rectangles intersecting a diagonal line: algorithms and complexity. Discret. Comput. Geom. 53(2), 344–365 (2015)
4. 4.
Das, G.K., De, M., Kolay, S., Nandy, S.C., Sur-Kolay, S.: Approximation algorithms for maximum independent set of a unit disk graph. Inf. Process. Lett. 115(3), 439–446 (2015)
5. 5.
Fraser, R., López-Ortiz, A.: The within-strip discrete unit disk cover problem. Theor. Comput. Sci. 674, 99–115 (2017)
6. 6.
Garey, M.R., Johnson, D.S.: The rectilinear steiner tree problem is NP-complete. SIAM J. Appl. Math. 32(4), 826–834 (1977)
7. 7.
Knuth, D.E., Raghunathan, A.: The problem of compatible representatives. SIAM J. Discret. Math. 5(3), 422–427 (1992)
8. 8.
Kratochvíl, J., Nešetřil, J.: INDEPENDENT SET and CLIQUE problems in intersection-defined classes of graphs. Commentationes Mathematicae Universitatis Carolinae 031(1), 85–93 (1990)
9. 9.
Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11(2), 329–343 (1982)
10. 10.
Lubiw, A.: A weighted min-max relation for intervals. J. Comb. Theory Ser. B 53(2), 151–172 (1991)
11. 11.
Mudgal, A., Pandit, S.: Covering, hitting, piercing and packing rectangles intersecting an inclined line. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, D.-Z. (eds.) COCOA 2015. LNCS, vol. 9486, pp. 126–137. Springer, Cham (2015).