# Watchman route problem

The Watchman Problem is an optimization problem in computational geometry where the objective is to compute the shortest route a watchman should take to guard an entire area with obstacles given only a map of the area. The challenge is to make sure the watchman peeks behind every corner and to determine the best order in which corners should be visited in. The problem may be solved in polynomial time when the area to be guarded is a simple polygon.[1][2][3] The problem is NP-hard for polygons with holes,[1] but may be approximated in polynomial time by a solution whose length is within a polylogarithmic factor of optimal.[4]

• Art gallery problem, which similarly involves viewing all points of a given area, but with multiple stationary watchmen

## References

1. ^ a b Chin, Wei-Pang; Ntafos, Simeon (1988), "Optimum watchman routes", Information Processing Letters 28 (1): 39–44, doi:10.1016/0020-0190(88)90141-X, MR 947253.
2. ^ Carlsson, S.; Jonsson, H.; Nilsson, B. J. (1999), "Finding the shortest watchman route in a simple polygon", Discrete and Computational Geometry 22 (3): 377–402, doi:10.1007/PL00009467, MR 1706598.
3. ^ Tan, Xuehou (2001), "Fast computation of shortest watchman routes in simple polygons", Information Processing Letters 77 (1): 27–33, doi:10.1016/S0020-0190(00)00146-0, MR 1813864.
4. ^ Mitchell, Joseph S. B. (2013), "Approximating watchman routes", Proceedings of the Twenty-Fourth Annual ACM–SIAM Symposium on Discrete Algorithms (SODA '13), SIAM, pp. 844–855, doi:10.1137/1.9781611973105.60, ISBN 978-1-611972-51-1.