Bellman's lost-in-a-forest problem

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Unsolved problem in mathematics:
What is the optimal path to take when lost in a forest?

Bellman's lost-in-a-forest problem is an unsolved minimization problem in geometry, originating in 1955 by the American applied mathematician Richard E. Bellman. [1] The problem is often stated as follows: "A hiker is lost in a forest whose shape and dimensions are precisely known to him. What is the best path for him to follow to escape from the forest?" [2] It is usually assumed that the hiker does not know the starting point or direction he is facing. The best path is taken to be the one that minimizes the worst-case distance to travel before reaching the edge of the forest. Other variations of the problem have been studied.

Although non-contrived real-world applications are not apparent, the problem falls into a class of geometric optimization problems, including search strategies that are of practical importance. A bigger motivation for study has been the connection to Moser's worm problem. It was included in a list of 12 problems described by the mathematician Scott W. Williams as "million buck problems" because he believed that the techniques involved in their resolution will be worth at least a million dollars to mathematics. [3]

Approaches

A proven solution is only known for a few shapes or classes of shape, such as regular polygons and a circle. In particular, all shapes which can enclose a 60° rhombus with longer diagonal equal to the diameter have a solution of a straight line. The equilateral triangle is the only regular polygon which does not have this property, and has a solution consisting of a zig-zag line with three segments of equal length. The solution for many other shapes remains unknown. [4] A general solution would be in the form of a geometric algorithm which takes the shape of the forest as input and returns the optimal escape path as the output.

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References

  1. Bellman, R. (1956). "Minimization problem". Research problems. Bulletin of the American Mathematical Society . 62 (3): 270. doi: 10.1090/S0002-9904-1956-10021-9 .
  2. Finch, S. R.; Wetzel, J. E. (2004). "Lost in a forest" (PDF). American Mathematical Monthly . 11 (8): 645–654. doi:10.2307/4145038. JSTOR   4145038. MR   2091541.
  3. Williams, S. W. (2000). "Million buck problems" (PDF). National Association of Mathematicians Newsletter. 31 (2): 1–3.
  4. Ward, John W. (2008). "Exploring the Bellman Forest Problem" (PDF). Retrieved 2020-12-14.