  The two-watchman route problem is that of computing a pair of closed tours in
an environment so that the two tours together see the whole environment and
some length measure on the two tours is minimized. Two standard measures are:
the minmax measure, where we want the tours where the longest of them has
smallest length, and the minsum measure, where we want the tours for which the
sum of their lengths is the smallest. It is known that computing a minmax
two-watchman route is NP-hard for simple rectilinear polygons and thus also for
simple polygons. Also, any c-approximation algorithm for the minmax
two-watchman route is automatically a 2c-approximation algorithm for the minsum
two-watchman route. We exhibit two constant factor approximation algorithms for
computing minmax two-watchman routes in simple polygons with approximation
factors 5.969 and 11.939, having running times O(n^8) and O(n^4) respectively,
where n is the number of vertices of the polygon. We also use the same
techniques to obtain a 6.922-approximation for the fixed two-watchman route
problem running in O(n^2) time, i.e., when two starting points of the two tours
are given as input.
