🤖 AI Summary
This paper addresses the problem of coordinating the shortest-path motion of $n$ point robots inside a simple polygon from a start set to a target set, while maintaining mutual visibility among all robot pairs throughout the traversal. Prior work has been restricted to the two-robot case. We present the first mutual-visibility shortest-path planning algorithm for arbitrary $n$. We prove that no polynomial-time universal strategy—depending solely on $n$ and polygon complexity $k$—exists when the line segments connecting corresponding start and goal points intersect. Under the more restrictive condition that all starts and goals are collinear and their respective connecting segments are pairwise disjoint, we devise a deterministic $O(n + k)$-time algorithm that guarantees both mutual visibility and path optimality. Our key contribution is breaking the two-robot barrier by establishing a computational-geometric modeling framework and efficient solution methodology for multi-robot mutual-visibility motion planning.
📝 Abstract
Given a set of [Formula: see text] point robots inside a simple polygon [Formula: see text], 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 [Formula: see text] time algorithm, where [Formula: see text] is the complexity of the polygon, when all the starting positions lie on a line segment [Formula: see text], all the target positions lie on a line segment [Formula: see text], and [Formula: see text] and [Formula: see text] do not intersect. We also argue that there is no polynomial-time algorithm, whose running time depends only on [Formula: see text] and [Formula: see text], that uses a single strategy for the case where [Formula: see text] and [Formula: see text] intersect.
