The Art Gallery Problem

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The Art Gallery Problem

• Presentation for MA 341
• Joseph Dewees
• December 1, 1999

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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?

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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?

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The Art Gallery Theorem

• In 1975, Vasek Chvatal solved Klees problem,
  using the following theorem
• n/3 guards are occasionally necessary and
  always sufficient to cover a polygon with n
  vertices.

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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 Fisks method of proving the theorem.

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Possible polygons (art galleries)
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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 polygons boundaries. There are many ways to
  do this.

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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.

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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.

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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.

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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.

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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?

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Sources

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