Multiple Watchman Routes in Staircase Polygons

TL;DR

This paper addresses the multiple watchman route problem in staircase polygons, proposing an optimal quadratic-time algorithm for two watchmen and an approximation for three or more, advancing computational efficiency in this geometric problem.

Contribution

It introduces a quadratic-time algorithm for two watchmen and an approximation method for three or more, improving efficiency over previous solutions.

Findings

Optimal quadratic-time algorithm for two watchmen

Approximation algorithm with additive error for three or more watchmen

Identification of limitations in extending the approach to more watchmen

Abstract

We consider the watchman route problem for multiple watchmen in staircase polygons, which are rectilinear $x$- and $y$-monotone polygons. For two watchmen, we propose an algorithm to find an optimal solution that takes quadratic time, improving on the cubic time of a trivial solution. For $m \geq 3$ watchmen, we explain where this approach fails, and present an approximation algorithm for the min-max criterion with only an additive error.