m-Watchmen's routes in minbar and generalized minbar polygons

Abstract

We study the problem of multiple anchored watchman routes, where we are given m starting points for watchmen, and aim to find routes for all watchmen such that all points in a polygon are visible from at least one route. We consider the problem in Minbar polygons,² which are staircase polygons for which the floor of the staircase solely consists of one horizontal and one vertical edge, and in generalized Minbar polygons, which relaxes the definition of Minbar polygons, allowing for non-rectilinear edges. For Minbar polygons, we exhibit polynomial time algorithms to compute optimal solutions for both the min-max and the min-sum criteria. The min-max algorithm takes $O(m\log m+n\log n)$ time, using $O(m+n)$ storage, and the min-sum algorithm takes $O(n^2\log m+m\log m)$ time, also using $O(m+n)$ storage.

For generalized Minbar polygons, we prove NP-hardness for the min-sum and min-max criteria, and present approximation algorithms for both criteria: an $O(\log (m+n\left)\right)$-approximation taking $O\left(m^4{n}^2\right)$ time for the min-sum criterion, and a $(\pi +3)$-approximation taking $O\left(m^3{n}^2\right)$ time for the min-max criterion.

Minbar polygons and the non-rectilinear generalization of them may seem to be very restricted polygon classes but they form an adjacent pair where the multiple anchored watchman routes problem has a polynomial time solution in one class but is NP-hard in the slightly more generalized class. It is this property that motivates our study of these restricted polygon classes.