Approximation Algorithms for the Two-Watchman Route in a Simple Polygon

TL;DR

This paper presents new constant-factor approximation algorithms for the two-watchman route problem in simple polygons, improving computational efficiency for finding near-optimal tours that cover the entire environment.

Contribution

The authors develop the first constant-factor approximation algorithms for minmax two-watchman routes in simple polygons with improved running times.

Findings

Achieved 5.969 and 11.939 approximation factors with algorithms running in O(n^8) and O(n^4) time.

Provided a 6.922-approximation algorithm for fixed start points with O(n^2) runtime.

Demonstrated the applicability of techniques to both minmax and fixed start scenarios.

Abstract

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.