The Art Gallery Problem

1. The Art Gallery Problem Presentation for MA 341 Joseph Dewees December 1, 1999

2. What is the art gallery problem? • You own an art gallery and want to place security cameras so that the entire gallery will be safe from theives. • Where should you place the cameras? • What is the minimum number of cameras you will need to keep you art collection safe?

3. History • In 1973 Victor Klee considered the following problem: Consider an art gallery whose floor plan can be modeled by a polygon with n vertices. • What is the minimum number of stationary guards needed to protect the room?

4. The Art Gallery Theorem • In 1975, Vasek Chvatal solved Klee’s problem, using the following theorem: • [n/3] guards are occasionally necessary and always sufficient to cover a polygon with n vertices.

5. Proofs • Chvatal constructed the first proof of his theorem in 1975. It was very elaborate and used induction. In 1978 Steve Fisk constructed a much simpler proof based on dividing a polygon into triangles using diagonals. We will concentrate on Fisk’s method of proving the theorem.

6. Possible polygons (art galleries)

7. Subdivide into triangles • First, we divide the polygon into triangles, with the vertices of the polygon becoming the vertices of the triangles. Some vertices may belong to more than one triangle. • We are careful to make sure that none of the lines we add cross one another or pass outside the polygon’s boundaries. There are many ways to do this.

8. Apply the Three-Color Theorem • Next, we apply a theorem which says that the vertices of any triangulated polygon can be three-colored. • Using only red, blue, and green, I can color all of the vertices of the polygon so that no two adjacent vertices are the same color. • If done correctly, each triangle will end up with one corner of each color.

9. Placing the guards • I can now pick one of the colors and put a guard at each corner having that color. • For a figure with n vertices, where n is not divisible by three, all colors will not have an equal number of vertices. We want to know the least number of guards we can use, so we choose a color with the least number of vertices.

10. Problem solved • Since each triangle has each color on its three vertices, we know that by placing the guards at the corners with one given color, the guards will be able to see each triangle, collectively. Since each triangle is protected, the entire polygon is protected. • We have shown that a polygon of n vertices can be guarded by [n/3] guards.

11. Variations • In 1980, Kahn, Klawe, and Kleitman proved that the number of guards necessary and sufficient to protect a rectilinear polygon with n vertices was [n/4]. • In 1982, Shermer examined a more realistic floor plan for an art gallery. This room had obstacles, which he represented with holes. He was able to solve the problem for n vertices and h holes.

12. Applications • Solutions to the Art Gallery problem have provided strategies for improving many security problems. • For example, where, on college campuses, are the best locations to place security officers and how many are needed?

13. Sources • I used the following internet sources for this presentation: • http://dimacs.rutgers.edu/drei/96/classroom/art/history.html • http://www.maa.org/mathland/mathland_11_4.html • http://www.cs.mcgill.ca/~thierry/artgallery.html