Shortest Paths of Mutually Visible Robots

TL;DR

This paper develops an efficient algorithm for moving multiple robots along shortest paths inside a polygon while maintaining mutual visibility, extending previous work from two robots to multiple robots under specific conditions.

Contribution

It introduces an $O(mn)$ time algorithm for multiple robots with start and target positions on non-intersecting line segments, and discusses the complexity when segments intersect.

Findings

Algorithm runs in $O(mn)$ time for non-intersecting segments.

No polynomial-time strategy exists for intersecting segments case.

Extends shortest path visibility preservation from two robots to multiple robots.

Abstract

Given a set of $n$ point robots inside a simple polygon $P$, the task is to move the robots from their starting positions to their target positions along their shortest paths, while the mutual visibility of these robots is preserved. Previous work only considered two robots. In this paper, we present an $O(mn)$ time algorithm, where $m$ is the complexity of the polygon, when all the starting positions lie on a line segment $S$, all the target positions lie on a line segment $T$, and $S$ and $T$ do not intersect. We also argue that there is no polynomial-time algorithm, whose running time depends only on $n$ and $m$, that uses a single strategy for the case where $S$ and $T$ intersect.